Theorem mzpcompact2lem 42767

Description: Lemma for mzpcompact2 42768. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Hypothesis

Ref	Expression
mzpcompact2lem.i	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
mzpcompact2lem	⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏,𝑐

Allowed substitution hint:   𝐴(𝑐)

Proof of Theorem mzpcompact2lem

Dummy variables 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1543	. . 3 ⊢ ⊤
2		0fi 9083	. . . . . 6 ⊢ ∅ ∈ Fin
3		0ex 5306	. . . . . . . 8 ⊢ ∅ ∈ V
4		mzpconst 42751	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝑓 ∈ ℤ) → ((ℤ ↑m ∅) × {𝑓}) ∈ (mzPoly‘∅))
5	3, 4	mpan 690	. . . . . . 7 ⊢ (𝑓 ∈ ℤ → ((ℤ ↑m ∅) × {𝑓}) ∈ (mzPoly‘∅))
6		0ss 4399	. . . . . . . 8 ⊢ ∅ ⊆ 𝐵
7	6	a1i 11	. . . . . . 7 ⊢ (𝑓 ∈ ℤ → ∅ ⊆ 𝐵)
8		fconstmpt 5746	. . . . . . . 8 ⊢ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ 𝑓)
9		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℤ ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → 𝑑 ∈ (ℤ ↑m 𝐵))
10		elmapssres 8908	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ∧ ∅ ⊆ 𝐵) → (𝑑 ↾ ∅) ∈ (ℤ ↑m ∅))
11	9, 6, 10	sylancl 586	. . . . . . . . . 10 ⊢ ((𝑓 ∈ ℤ ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (𝑑 ↾ ∅) ∈ (ℤ ↑m ∅))
12		vex 3483	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
13	12	fvconst2 7225	. . . . . . . . . 10 ⊢ ((𝑑 ↾ ∅) ∈ (ℤ ↑m ∅) → (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅)) = 𝑓)
14	11, 13	syl 17	. . . . . . . . 9 ⊢ ((𝑓 ∈ ℤ ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅)) = 𝑓)
15	14	mpteq2dva 5241	. . . . . . . 8 ⊢ (𝑓 ∈ ℤ → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ 𝑓))
16	8, 15	eqtr4id 2795	. . . . . . 7 ⊢ (𝑓 ∈ ℤ → ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅))))
17		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑏 = ((ℤ ↑m ∅) × {𝑓}) → (𝑏‘(𝑑 ↾ ∅)) = (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅)))
18	17	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑏 = ((ℤ ↑m ∅) × {𝑓}) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅))))
19	18	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = ((ℤ ↑m ∅) × {𝑓}) → (((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))) ↔ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅)))))
20	19	anbi2d 630	. . . . . . . 8 ⊢ (𝑏 = ((ℤ ↑m ∅) × {𝑓}) → ((∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅)))) ↔ (∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅))))))
21	20	rspcev 3621	. . . . . . 7 ⊢ ((((ℤ ↑m ∅) × {𝑓}) ∈ (mzPoly‘∅) ∧ (∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (((ℤ ↑m ∅) × {𝑓})‘(𝑑 ↾ ∅))))) → ∃𝑏 ∈ (mzPoly‘∅)(∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅)))))
22	5, 7, 16, 21	syl12anc 836	. . . . . 6 ⊢ (𝑓 ∈ ℤ → ∃𝑏 ∈ (mzPoly‘∅)(∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅)))))
23		fveq2 6905	. . . . . . . 8 ⊢ (𝑎 = ∅ → (mzPoly‘𝑎) = (mzPoly‘∅))
24		sseq1 4008	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵))
25		reseq2 5991	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑑 ↾ 𝑎) = (𝑑 ↾ ∅))
26	25	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑑 ↾ ∅)))
27	26	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))))
28	27	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅)))))
29	24, 28	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))))))
30	23, 29	rexeqbidv 3346	. . . . . . 7 ⊢ (𝑎 = ∅ → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘∅)(∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))))))
31	30	rspcev 3621	. . . . . 6 ⊢ ((∅ ∈ Fin ∧ ∃𝑏 ∈ (mzPoly‘∅)(∅ ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ∅))))) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
32	2, 22, 31	sylancr 587	. . . . 5 ⊢ (𝑓 ∈ ℤ → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
33	32	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑓 ∈ ℤ) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
34		snfi 9084	. . . . . 6 ⊢ {𝑓} ∈ Fin
35		vsnex 5433	. . . . . . . . 9 ⊢ {𝑓} ∈ V
36		vsnid 4662	. . . . . . . . 9 ⊢ 𝑓 ∈ {𝑓}
37		mzpproj 42753	. . . . . . . . 9 ⊢ (({𝑓} ∈ V ∧ 𝑓 ∈ {𝑓}) → (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) ∈ (mzPoly‘{𝑓}))
38	35, 36, 37	mp2an 692	. . . . . . . 8 ⊢ (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) ∈ (mzPoly‘{𝑓})
39	38	a1i 11	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 → (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) ∈ (mzPoly‘{𝑓}))
40		snssi 4807	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 → {𝑓} ⊆ 𝐵)
41		fveq1 6904	. . . . . . . . 9 ⊢ (𝑔 = 𝑑 → (𝑔‘𝑓) = (𝑑‘𝑓))
42	41	cbvmptv 5254	. . . . . . . 8 ⊢ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑑‘𝑓))
43		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → 𝑑 ∈ (ℤ ↑m 𝐵))
44		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → 𝑓 ∈ 𝐵)
45	44	snssd 4808	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → {𝑓} ⊆ 𝐵)
46		elmapssres 8908	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ∧ {𝑓} ⊆ 𝐵) → (𝑑 ↾ {𝑓}) ∈ (ℤ ↑m {𝑓}))
47	43, 45, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (𝑑 ↾ {𝑓}) ∈ (ℤ ↑m {𝑓}))
48		fveq1 6904	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑑 ↾ {𝑓}) → (𝑔‘𝑓) = ((𝑑 ↾ {𝑓})‘𝑓))
49		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) = (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))
50		fvex 6918	. . . . . . . . . . . 12 ⊢ ((𝑑 ↾ {𝑓})‘𝑓) ∈ V
51	48, 49, 50	fvmpt 7015	. . . . . . . . . . 11 ⊢ ((𝑑 ↾ {𝑓}) ∈ (ℤ ↑m {𝑓}) → ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓})) = ((𝑑 ↾ {𝑓})‘𝑓))
52	47, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓})) = ((𝑑 ↾ {𝑓})‘𝑓))
53		fvres 6924	. . . . . . . . . . 11 ⊢ (𝑓 ∈ {𝑓} → ((𝑑 ↾ {𝑓})‘𝑓) = (𝑑‘𝑓))
54	36, 53	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑑 ↾ {𝑓})‘𝑓) = (𝑑‘𝑓)
55	52, 54	eqtr2di 2793	. . . . . . . . 9 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (𝑑‘𝑓) = ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓})))
56	55	mpteq2dva 5241	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐵 → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑑‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓}))))
57	42, 56	eqtrid 2788	. . . . . . 7 ⊢ (𝑓 ∈ 𝐵 → (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓}))))
58		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) → (𝑏‘(𝑑 ↾ {𝑓})) = ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓})))
59	58	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑏 = (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓}))))
60	59	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) → ((𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))) ↔ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓})))))
61	60	anbi2d 630	. . . . . . . 8 ⊢ (𝑏 = (𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) → (({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓})))) ↔ ({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓}))))))
62	61	rspcev 3621	. . . . . . 7 ⊢ (((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓)) ∈ (mzPoly‘{𝑓}) ∧ ({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑔 ∈ (ℤ ↑m {𝑓}) ↦ (𝑔‘𝑓))‘(𝑑 ↾ {𝑓}))))) → ∃𝑏 ∈ (mzPoly‘{𝑓})({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓})))))
63	39, 40, 57, 62	syl12anc 836	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 → ∃𝑏 ∈ (mzPoly‘{𝑓})({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓})))))
64		fveq2 6905	. . . . . . . 8 ⊢ (𝑎 = {𝑓} → (mzPoly‘𝑎) = (mzPoly‘{𝑓}))
65		sseq1 4008	. . . . . . . . 9 ⊢ (𝑎 = {𝑓} → (𝑎 ⊆ 𝐵 ↔ {𝑓} ⊆ 𝐵))
66		reseq2 5991	. . . . . . . . . . . 12 ⊢ (𝑎 = {𝑓} → (𝑑 ↾ 𝑎) = (𝑑 ↾ {𝑓}))
67	66	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑎 = {𝑓} → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑑 ↾ {𝑓})))
68	67	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑎 = {𝑓} → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))))
69	68	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = {𝑓} → ((𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓})))))
70	65, 69	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = {𝑓} → ((𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))))))
71	64, 70	rexeqbidv 3346	. . . . . . 7 ⊢ (𝑎 = {𝑓} → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘{𝑓})({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))))))
72	71	rspcev 3621	. . . . . 6 ⊢ (({𝑓} ∈ Fin ∧ ∃𝑏 ∈ (mzPoly‘{𝑓})({𝑓} ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ {𝑓}))))) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
73	34, 63, 72	sylancr 587	. . . . 5 ⊢ (𝑓 ∈ 𝐵 → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
74	73	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑓 ∈ 𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
75		simplll 774	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ℎ ∈ Fin)
76		simprll 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝑗 ∈ Fin)
77		unfi 9212	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ ∈ Fin ∧ 𝑗 ∈ Fin) → (ℎ ∪ 𝑗) ∈ Fin)
78	75, 76, 77	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (ℎ ∪ 𝑗) ∈ Fin)
79		vex 3483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℎ ∈ V
80		vex 3483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑗 ∈ V
81	79, 80	unex 7765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∪ 𝑗) ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (ℎ ∪ 𝑗) ∈ V)
83		ssun1 4177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℎ ⊆ (ℎ ∪ 𝑗)
84	83	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ℎ ⊆ (ℎ ∪ 𝑗))
85		simpllr 775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝑖 ∈ (mzPoly‘ℎ))
86		mzpresrename 42766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ ∪ 𝑗) ∈ V ∧ ℎ ⊆ (ℎ ∪ 𝑗) ∧ 𝑖 ∈ (mzPoly‘ℎ)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑖‘(𝑙 ↾ ℎ))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
87	82, 84, 85, 86	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑖‘(𝑙 ↾ ℎ))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
88		ssun2 4178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑗 ⊆ (ℎ ∪ 𝑗)
89	88	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝑗 ⊆ (ℎ ∪ 𝑗))
90		simprlr 779	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝑘 ∈ (mzPoly‘𝑗))
91		mzpresrename 42766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ ∪ 𝑗) ∈ V ∧ 𝑗 ⊆ (ℎ ∪ 𝑗) ∧ 𝑘 ∈ (mzPoly‘𝑗)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑘‘(𝑙 ↾ 𝑗))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
92	82, 89, 90, 91	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑘‘(𝑙 ↾ 𝑗))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
93		mzpaddmpt 42757	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑖‘(𝑙 ↾ ℎ))) ∈ (mzPoly‘(ℎ ∪ 𝑗)) ∧ (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑘‘(𝑙 ↾ 𝑗))) ∈ (mzPoly‘(ℎ ∪ 𝑗))) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
94	87, 92, 93	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
95		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ℎ ⊆ 𝐵)
96		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝑗 ⊆ 𝐵)
97	95, 96	unssd 4191	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (ℎ ∪ 𝑗) ⊆ 𝐵)
98		ovex 7465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ ↑m 𝐵) ∈ V
99	98	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (ℤ ↑m 𝐵) ∈ V)
100		mzpcompact2lem.i	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐵 ∈ V
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → 𝐵 ∈ V)
102		mzpresrename 42766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ V ∧ ℎ ⊆ 𝐵 ∧ 𝑖 ∈ (mzPoly‘ℎ)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∈ (mzPoly‘𝐵))
103	101, 95, 85, 102	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∈ (mzPoly‘𝐵))
104		mzpf 42752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∈ (mzPoly‘𝐵) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))):(ℤ ↑m 𝐵)⟶ℤ)
105		ffn 6735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))):(ℤ ↑m 𝐵)⟶ℤ → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) Fn (ℤ ↑m 𝐵))
106	103, 104, 105	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) Fn (ℤ ↑m 𝐵))
107		mzpresrename 42766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ V ∧ 𝑗 ⊆ 𝐵 ∧ 𝑘 ∈ (mzPoly‘𝑗)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) ∈ (mzPoly‘𝐵))
108	101, 96, 90, 107	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) ∈ (mzPoly‘𝐵))
109		mzpf 42752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) ∈ (mzPoly‘𝐵) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))):(ℤ ↑m 𝐵)⟶ℤ)
110		ffn 6735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))):(ℤ ↑m 𝐵)⟶ℤ → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) Fn (ℤ ↑m 𝐵))
111	108, 109, 110	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) Fn (ℤ ↑m 𝐵))
112		ofmpteq 7721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℤ ↑m 𝐵) ∈ V ∧ (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) Fn (ℤ ↑m 𝐵) ∧ (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) Fn (ℤ ↑m 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) + (𝑘‘(𝑑 ↾ 𝑗)))))
113	99, 106, 111, 112	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) + (𝑘‘(𝑑 ↾ 𝑗)))))
114		elmapi 8890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑑 ∈ (ℤ ↑m 𝐵) → 𝑑:𝐵⟶ℤ)
115		fssres 6773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑑:𝐵⟶ℤ ∧ (ℎ ∪ 𝑗) ⊆ 𝐵) → (𝑑 ↾ (ℎ ∪ 𝑗)):(ℎ ∪ 𝑗)⟶ℤ)
116	114, 97, 115	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (𝑑 ↾ (ℎ ∪ 𝑗)):(ℎ ∪ 𝑗)⟶ℤ)
117		zex 12624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℤ ∈ V
118	117, 81	elmap 8912	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ↾ (ℎ ∪ 𝑗)) ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↔ (𝑑 ↾ (ℎ ∪ 𝑗)):(ℎ ∪ 𝑗)⟶ℤ)
119	116, 118	sylibr 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → (𝑑 ↾ (ℎ ∪ 𝑗)) ∈ (ℤ ↑m (ℎ ∪ 𝑗)))
120		reseq1 5990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → (𝑙 ↾ ℎ) = ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ))
121	120	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → (𝑖‘(𝑙 ↾ ℎ)) = (𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)))
122		reseq1 5990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → (𝑙 ↾ 𝑗) = ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))
123	122	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → (𝑘‘(𝑙 ↾ 𝑗)) = (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗)))
124	121, 123	oveq12d 7450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) + (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
125		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))
126		ovex 7465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) + (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))) ∈ V
127	124, 125, 126	fvmpt 7015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ↾ (ℎ ∪ 𝑗)) ∈ (ℤ ↑m (ℎ ∪ 𝑗)) → ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) + (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
128	119, 127	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) + (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
129		resabs1 6023	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ ⊆ (ℎ ∪ 𝑗) → ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ) = (𝑑 ↾ ℎ))
130	83, 129	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ) = (𝑑 ↾ ℎ)
131	130	fveq2i 6908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) = (𝑖‘(𝑑 ↾ ℎ))
132		resabs1 6023	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ⊆ (ℎ ∪ 𝑗) → ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗) = (𝑑 ↾ 𝑗))
133	88, 132	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗) = (𝑑 ↾ 𝑗)
134	133	fveq2i 6908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗)) = (𝑘‘(𝑑 ↾ 𝑗))
135	131, 134	oveq12i 7444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) + (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))) = ((𝑖‘(𝑑 ↾ ℎ)) + (𝑘‘(𝑑 ↾ 𝑗)))
136	128, 135	eqtr2di 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → ((𝑖‘(𝑑 ↾ ℎ)) + (𝑘‘(𝑑 ↾ 𝑗))) = ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))
137	136	mpteq2dva 5241	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) + (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
138	113, 137	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
139		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) → (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))
140	139	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
141	140	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) → (((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
142	141	anbi2d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) → (((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))) ↔ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
143	142	rspcev 3621	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)) ∧ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) + (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))) → ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
144	94, 97, 138, 143	syl12anc 836	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
145		mzpmulmpt 42758	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑖‘(𝑙 ↾ ℎ))) ∈ (mzPoly‘(ℎ ∪ 𝑗)) ∧ (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ (𝑘‘(𝑙 ↾ 𝑗))) ∈ (mzPoly‘(ℎ ∪ 𝑗))) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
146	87, 92, 145	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)))
147		ofmpteq 7721	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℤ ↑m 𝐵) ∈ V ∧ (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) Fn (ℤ ↑m 𝐵) ∧ (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))) Fn (ℤ ↑m 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) · (𝑘‘(𝑑 ↾ 𝑗)))))
148	99, 106, 111, 147	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) · (𝑘‘(𝑑 ↾ 𝑗)))))
149	121, 123	oveq12d 7450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = (𝑑 ↾ (ℎ ∪ 𝑗)) → ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) · (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
150		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))
151		ovex 7465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) · (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))) ∈ V
152	149, 150, 151	fvmpt 7015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑑 ↾ (ℎ ∪ 𝑗)) ∈ (ℤ ↑m (ℎ ∪ 𝑗)) → ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) · (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
153	119, 152	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) · (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))))
154	131, 134	oveq12i 7444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ ℎ)) · (𝑘‘((𝑑 ↾ (ℎ ∪ 𝑗)) ↾ 𝑗))) = ((𝑖‘(𝑑 ↾ ℎ)) · (𝑘‘(𝑑 ↾ 𝑗)))
155	153, 154	eqtr2di 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) ∧ 𝑑 ∈ (ℤ ↑m 𝐵)) → ((𝑖‘(𝑑 ↾ ℎ)) · (𝑘‘(𝑑 ↾ 𝑗))) = ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))
156	155	mpteq2dva 5241	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑖‘(𝑑 ↾ ℎ)) · (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
157	148, 156	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
158		fveq1 6904	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) → (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))) = ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))
159	158	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
160	159	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) → (((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
161	160	anbi2d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 = (𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) → (((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))) ↔ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
162	161	rspcev 3621	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗)))) ∈ (mzPoly‘(ℎ ∪ 𝑗)) ∧ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ ((𝑙 ∈ (ℤ ↑m (ℎ ∪ 𝑗)) ↦ ((𝑖‘(𝑙 ↾ ℎ)) · (𝑘‘(𝑙 ↾ 𝑗))))‘(𝑑 ↾ (ℎ ∪ 𝑗)))))) → ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
163	146, 97, 157, 162	syl12anc 836	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
164		fveq2 6905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (mzPoly‘𝑎) = (mzPoly‘(ℎ ∪ 𝑗)))
165		sseq1 4008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (𝑎 ⊆ 𝐵 ↔ (ℎ ∪ 𝑗) ⊆ 𝐵))
166		reseq2 5991	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (𝑑 ↾ 𝑎) = (𝑑 ↾ (ℎ ∪ 𝑗)))
167	166	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))
168	167	mpteq2dv 5243	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))))
169	168	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
170	165, 169	anbi12d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (ℎ ∪ 𝑗) → ((𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
171	164, 170	rexeqbidv 3346	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
172	168	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))
173	165, 172	anbi12d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (ℎ ∪ 𝑗) → ((𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
174	164, 173	rexeqbidv 3346	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (ℎ ∪ 𝑗) → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗)))))))
175	171, 174	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (ℎ ∪ 𝑗) → ((∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))) ↔ (∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))) ∧ ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))))
176	175	rspcev 3621	. . . . . . . . . . . . . . . . 17 ⊢ (((ℎ ∪ 𝑗) ∈ Fin ∧ (∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))) ∧ ∃𝑏 ∈ (mzPoly‘(ℎ ∪ 𝑗))((ℎ ∪ 𝑗) ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ (ℎ ∪ 𝑗))))))) → ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
177	78, 144, 163, 176	syl12anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ ℎ ⊆ 𝐵) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
178	177	adantlrr 721	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ 𝑗 ⊆ 𝐵)) → ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
179	178	adantrrr 725	. . . . . . . . . . . . . 14 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
180		simplrr 777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))
181		simprrr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))
182	180, 181	oveq12d 7450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (𝑓 ∘f + 𝑔) = ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))
183	182	eqeq1d 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ((𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
184	183	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ((𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
185	184	rexbidv 3178	. . . . . . . . . . . . . . . 16 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
186	180, 181	oveq12d 7450	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (𝑓 ∘f · 𝑔) = ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))
187	186	eqeq1d 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ((𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
188	187	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ((𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
189	188	rexbidv 3178	. . . . . . . . . . . . . . . 16 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
190	185, 189	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ((∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))) ↔ (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))))
191	190	rexbidv 3178	. . . . . . . . . . . . . 14 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))) ↔ ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f + (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))) ∘f · (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))))
192	179, 191	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
193		r19.40 3118	. . . . . . . . . . . . 13 ⊢ (∃𝑎 ∈ Fin (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
194	192, 193	syl 17	. . . . . . . . . . . 12 ⊢ ((((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) ∧ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
195	194	exp32 420	. . . . . . . . . . 11 ⊢ (((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) → ((𝑗 ∈ Fin ∧ 𝑘 ∈ (mzPoly‘𝑗)) → ((𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))))
196	195	rexlimdvv 3211	. . . . . . . . . 10 ⊢ (((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) ∧ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) → (∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))))
197	196	ex 412	. . . . . . . . 9 ⊢ ((ℎ ∈ Fin ∧ 𝑖 ∈ (mzPoly‘ℎ)) → ((ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))) → (∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))))
198	197	rexlimivv 3200	. . . . . . . 8 ⊢ (∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))) → (∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))))
199	198	imp 406	. . . . . . 7 ⊢ ((∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))) ∧ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
200	199	ad2ant2l 746	. . . . . 6 ⊢ (((𝑓:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ (𝑔:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
201	200	3adant1 1130	. . . . 5 ⊢ ((⊤ ∧ (𝑓:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ (𝑔:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ∧ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
202	201	simpld 494	. . . 4 ⊢ ((⊤ ∧ (𝑓:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ (𝑔:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
203	201	simprd 495	. . . 4 ⊢ ((⊤ ∧ (𝑓:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))) ∧ (𝑔:(ℤ ↑m 𝐵)⟶ℤ ∧ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
204		eqeq1 2740	. . . . . 6 ⊢ (𝑒 = ((ℤ ↑m 𝐵) × {𝑓}) → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
205	204	anbi2d 630	. . . . 5 ⊢ (𝑒 = ((ℤ ↑m 𝐵) × {𝑓}) → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
206	205	2rexbidv 3221	. . . 4 ⊢ (𝑒 = ((ℤ ↑m 𝐵) × {𝑓}) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ ((ℤ ↑m 𝐵) × {𝑓}) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
207		eqeq1 2740	. . . . . 6 ⊢ (𝑒 = (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
208	207	anbi2d 630	. . . . 5 ⊢ (𝑒 = (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
209	208	2rexbidv 3221	. . . 4 ⊢ (𝑒 = (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑔 ∈ (ℤ ↑m 𝐵) ↦ (𝑔‘𝑓)) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
210		eqeq1 2740	. . . . . . 7 ⊢ (𝑒 = 𝑓 → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
211	210	anbi2d 630	. . . . . 6 ⊢ (𝑒 = 𝑓 → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
212	211	2rexbidv 3221	. . . . 5 ⊢ (𝑒 = 𝑓 → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
213		fveq2 6905	. . . . . . . 8 ⊢ (𝑎 = ℎ → (mzPoly‘𝑎) = (mzPoly‘ℎ))
214		sseq1 4008	. . . . . . . . 9 ⊢ (𝑎 = ℎ → (𝑎 ⊆ 𝐵 ↔ ℎ ⊆ 𝐵))
215		reseq2 5991	. . . . . . . . . . . 12 ⊢ (𝑎 = ℎ → (𝑑 ↾ 𝑎) = (𝑑 ↾ ℎ))
216	215	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑎 = ℎ → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑑 ↾ ℎ)))
217	216	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑎 = ℎ → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ))))
218	217	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = ℎ → (𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ)))))
219	214, 218	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = ℎ → ((𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ))))))
220	213, 219	rexeqbidv 3346	. . . . . . 7 ⊢ (𝑎 = ℎ → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ))))))
221		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑖 → (𝑏‘(𝑑 ↾ ℎ)) = (𝑖‘(𝑑 ↾ ℎ)))
222	221	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑏 = 𝑖 → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))
223	222	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = 𝑖 → (𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ))) ↔ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))))
224	223	anbi2d 630	. . . . . . . 8 ⊢ (𝑏 = 𝑖 → ((ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ)))) ↔ (ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))))
225	224	cbvrexvw 3237	. . . . . . 7 ⊢ (∃𝑏 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ ℎ)))) ↔ ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))))
226	220, 225	bitrdi 287	. . . . . 6 ⊢ (𝑎 = ℎ → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))))
227	226	cbvrexvw 3237	. . . . 5 ⊢ (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ)))))
228	212, 227	bitrdi 287	. . . 4 ⊢ (𝑒 = 𝑓 → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃ℎ ∈ Fin ∃𝑖 ∈ (mzPoly‘ℎ)(ℎ ⊆ 𝐵 ∧ 𝑓 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑖‘(𝑑 ↾ ℎ))))))
229		eqeq1 2740	. . . . . . 7 ⊢ (𝑒 = 𝑔 → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
230	229	anbi2d 630	. . . . . 6 ⊢ (𝑒 = 𝑔 → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
231	230	2rexbidv 3221	. . . . 5 ⊢ (𝑒 = 𝑔 → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
232		fveq2 6905	. . . . . . . 8 ⊢ (𝑎 = 𝑗 → (mzPoly‘𝑎) = (mzPoly‘𝑗))
233		sseq1 4008	. . . . . . . . 9 ⊢ (𝑎 = 𝑗 → (𝑎 ⊆ 𝐵 ↔ 𝑗 ⊆ 𝐵))
234		reseq2 5991	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑗 → (𝑑 ↾ 𝑎) = (𝑑 ↾ 𝑗))
235	234	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑗 → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑑 ↾ 𝑗)))
236	235	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑎 = 𝑗 → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗))))
237	236	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = 𝑗 → (𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗)))))
238	233, 237	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = 𝑗 → ((𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗))))))
239	232, 238	rexeqbidv 3346	. . . . . . 7 ⊢ (𝑎 = 𝑗 → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑏 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗))))))
240		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑘 → (𝑏‘(𝑑 ↾ 𝑗)) = (𝑘‘(𝑑 ↾ 𝑗)))
241	240	mpteq2dv 5243	. . . . . . . . . 10 ⊢ (𝑏 = 𝑘 → (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗))) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))
242	241	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = 𝑘 → (𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗))) ↔ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))
243	242	anbi2d 630	. . . . . . . 8 ⊢ (𝑏 = 𝑘 → ((𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗)))) ↔ (𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))))
244	243	cbvrexvw 3237	. . . . . . 7 ⊢ (∃𝑏 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑗)))) ↔ ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))
245	239, 244	bitrdi 287	. . . . . 6 ⊢ (𝑎 = 𝑗 → (∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))))
246	245	cbvrexvw 3237	. . . . 5 ⊢ (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗)))))
247	231, 246	bitrdi 287	. . . 4 ⊢ (𝑒 = 𝑔 → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑗 ∈ Fin ∃𝑘 ∈ (mzPoly‘𝑗)(𝑗 ⊆ 𝐵 ∧ 𝑔 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑘‘(𝑑 ↾ 𝑗))))))
248		eqeq1 2740	. . . . . 6 ⊢ (𝑒 = (𝑓 ∘f + 𝑔) → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
249	248	anbi2d 630	. . . . 5 ⊢ (𝑒 = (𝑓 ∘f + 𝑔) → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
250	249	2rexbidv 3221	. . . 4 ⊢ (𝑒 = (𝑓 ∘f + 𝑔) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f + 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
251		eqeq1 2740	. . . . . 6 ⊢ (𝑒 = (𝑓 ∘f · 𝑔) → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
252	251	anbi2d 630	. . . . 5 ⊢ (𝑒 = (𝑓 ∘f · 𝑔) → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
253	252	2rexbidv 3221	. . . 4 ⊢ (𝑒 = (𝑓 ∘f · 𝑔) → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ (𝑓 ∘f · 𝑔) = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
254		eqeq1 2740	. . . . . 6 ⊢ (𝑒 = 𝐴 → (𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
255	254	anbi2d 630	. . . . 5 ⊢ (𝑒 = 𝐴 → ((𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
256	255	2rexbidv 3221	. . . 4 ⊢ (𝑒 = 𝐴 → (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝑒 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))))))
257	33, 74, 202, 203, 206, 209, 228, 247, 250, 253, 256	mzpindd 42762	. . 3 ⊢ ((⊤ ∧ 𝐴 ∈ (mzPoly‘𝐵)) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
258	1, 257	mpan 690	. 2 ⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))))
259		reseq1 5990	. . . . . . 7 ⊢ (𝑑 = 𝑐 → (𝑑 ↾ 𝑎) = (𝑐 ↾ 𝑎))
260	259	fveq2d 6909	. . . . . 6 ⊢ (𝑑 = 𝑐 → (𝑏‘(𝑑 ↾ 𝑎)) = (𝑏‘(𝑐 ↾ 𝑎)))
261	260	cbvmptv 5254	. . . . 5 ⊢ (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))
262	261	eqeq2i 2749	. . . 4 ⊢ (𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎))) ↔ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎))))
263	262	anbi2i 623	. . 3 ⊢ ((𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ (𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))
264	263	2rexbii 3128	. 2 ⊢ (∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑑 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑑 ↾ 𝑎)))) ↔ ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))
265	258, 264	sylib 218	1 ⊢ (𝐴 ∈ (mzPoly‘𝐵) → ∃𝑎 ∈ Fin ∃𝑏 ∈ (mzPoly‘𝑎)(𝑎 ⊆ 𝐵 ∧ 𝐴 = (𝑐 ∈ (ℤ ↑m 𝐵) ↦ (𝑏‘(𝑐 ↾ 𝑎)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ⊤wtru 1540   ∈ wcel 2107  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  {csn 4625   ↦ cmpt 5224   × cxp 5682   ↾ cres 5686   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∘_f cof 7696   ↑_m cmap 8867  Fincfn 8986   + caddc 11159   · cmul 11161  ℤcz 12615  mzPolycmzp 42738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-mzpcl 42739  df-mzp 42740

This theorem is referenced by:  mzpcompact2  42768