Theorem pf1ind 20518

Description: Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
pf1ind.cb	⊢ 𝐵 = (Base‘𝑅)
pf1ind.cp	⊢ + = (+g‘𝑅)
pf1ind.ct	⊢ · = (.r‘𝑅)
pf1ind.cq	⊢ 𝑄 = ran (eval1‘𝑅)
pf1ind.ad	⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
pf1ind.mu	⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
pf1ind.wa	⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
pf1ind.wb	⊢ (𝑥 = ( I ↾ 𝐵) → (𝜓 ↔ 𝜃))
pf1ind.wc	⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
pf1ind.wd	⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
pf1ind.we	⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
pf1ind.wf	⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
pf1ind.wg	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
pf1ind.co	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
pf1ind.pr	⊢ (𝜑 → 𝜃)
pf1ind.a	⊢ (𝜑 → 𝐴 ∈ 𝑄)

Assertion

Ref	Expression
pf1ind	⊢ (𝜑 → 𝜌)

Distinct variable groups:   𝑓,𝑔,𝑥, +   𝐵,𝑓,𝑔,𝑥   𝜂,𝑓,𝑥   𝜑,𝑓,𝑔   𝑥,𝐴   𝜒,𝑥   𝜓,𝑓,𝑔   𝑄,𝑓,𝑔   𝜌,𝑥   𝜎,𝑥   𝜏,𝑥   𝜃,𝑥   · ,𝑓,𝑔,𝑥   𝜁,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑓,𝑔)   𝜃(𝑓,𝑔)   𝜏(𝑓,𝑔)   𝜂(𝑔)   𝜁(𝑓,𝑔)   𝜎(𝑓,𝑔)   𝜌(𝑓,𝑔)   𝐴(𝑓,𝑔)   𝑄(𝑥)   𝑅(𝑥,𝑓,𝑔)

Proof of Theorem pf1ind

Dummy variables 𝑎 𝑏 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coass 6118	. . . . 5 ⊢ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
2		df1o2 8116	. . . . . . . . 9 ⊢ 1o = {∅}
3		pf1ind.cb	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6684	. . . . . . . . 9 ⊢ 𝐵 ∈ V
5		0ex 5211	. . . . . . . . 9 ⊢ ∅ ∈ V
6		eqid 2821	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))
7	2, 4, 5, 6	mapsncnv 8457	. . . . . . . 8 ⊢ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
8	7	coeq2i 5731	. . . . . . 7 ⊢ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))
9	2, 4, 5, 6	mapsnf1o2 8458	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
10		f1ococnv2 6641	. . . . . . . 8 ⊢ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
11	9, 10	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
12	8, 11	syl5eqr 2870	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
13	12	coeq2d 5733	. . . . 5 ⊢ (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
14	1, 13	syl5eq 2868	. . . 4 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ( I ↾ 𝐵)))
15		pf1ind.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑄)
16		pf1ind.cq	. . . . . 6 ⊢ 𝑄 = ran (eval1‘𝑅)
17	16, 3	pf1f 20513	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → 𝐴:𝐵⟶𝐵)
18		fcoi1 6552	. . . . 5 ⊢ (𝐴:𝐵⟶𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
19	15, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
20	14, 19	eqtrd 2856	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = 𝐴)
21		pf1ind.cp	. . . 4 ⊢ + = (+g‘𝑅)
22		pf1ind.ct	. . . 4 ⊢ · = (.r‘𝑅)
23		eqid 2821	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
24	23, 3	evlval 20308	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
25	24	rneqi 5807	. . . 4 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26		an4 654	. . . . . 6 ⊢ (((𝑎 ∈ ran (1o eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1o eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) ↔ ((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})))
27		eqid 2821	. . . . . . . . . . . 12 ⊢ ran (1o eval 𝑅) = ran (1o eval 𝑅)
28	16, 3, 27	mpfpf1 20514	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
29	16, 3, 27	mpfpf1 20514	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
31		pf1ind.wc	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
32	30, 31	elab 3667	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
33		eleq1 2900	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
34	32, 33	syl5bbr 287	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
35	34	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝜏 ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂)))
36	35	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝜏 ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑)))
37		ovex 7189	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
38		pf1ind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
39	37, 38	elab 3667	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
40		oveq1 7163	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
41	40	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
42	39, 41	syl5bbr 287	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
43	36, 42	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓})))
44		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑔 ∈ V
45		pf1ind.wd	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
46	44, 45	elab 3667	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
47		eleq1 2900	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
48	46, 47	syl5bbr 287	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
49	48	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})))
50	49	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑)))
51		oveq2 7164	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))))
52	51	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
53	50, 52	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
54		pf1ind.ad	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
55	54	expcom 416	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜁))
56	55	an4s 658	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜁))
57	56	expimpd 456	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁))
58	43, 53, 57	vtocl2ga 3575	. . . . . . . . . . 11 ⊢ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
59	28, 29, 58	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
60	59	expcomd 419	. . . . . . . . 9 ⊢ ((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
61	60	impcom 410	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
62	25, 3	mpff 20317	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
63	62	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
64	63	ffnd 6515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵 ↑m 1o))
65	25, 3	mpff 20317	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
66	65	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
67	66	ffnd 6515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵 ↑m 1o))
68		eqid 2821	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
69	2, 4, 5, 68	mapsnf1o3 8459	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵–1-1-onto→(𝐵 ↑m 1o)
70		f1of 6615	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵–1-1-onto→(𝐵 ↑m 1o) → (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵⟶(𝐵 ↑m 1o))
71	69, 70	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵⟶(𝐵 ↑m 1o))
72		ovexd 7191	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (𝐵 ↑m 1o) ∈ V)
73	4	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝐵 ∈ V)
74		inidm 4195	. . . . . . . . . 10 ⊢ ((𝐵 ↑m 1o) ∩ (𝐵 ↑m 1o)) = (𝐵 ↑m 1o)
75	64, 67, 71, 72, 72, 73, 74	ofco 7429	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))))
76	75	eleq1d 2897	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
77	61, 76	sylibrd 261	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
78	77	expimpd 456	. . . . . 6 ⊢ (𝜑 → (((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
79	26, 78	syl5bi 244	. . . . 5 ⊢ (𝜑 → (((𝑎 ∈ ran (1o eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1o eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
80	79	imp 409	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran (1o eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1o eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))) → ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
81		ovex 7189	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
82		pf1ind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
83	81, 82	elab 3667	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
84		oveq1 7163	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
85	84	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
86	83, 85	syl5bbr 287	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
87	36, 86	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓})))
88		oveq2 7164	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))))
89	88	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
90	50, 89	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
91		pf1ind.mu	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
92	91	expcom 416	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜎))
93	92	an4s 658	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜎))
94	93	expimpd 456	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎))
95	87, 90, 94	vtocl2ga 3575	. . . . . . . . . . 11 ⊢ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
96	28, 29, 95	syl2an 597	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
97	96	expcomd 419	. . . . . . . . 9 ⊢ ((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
98	97	impcom 410	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
99	64, 67, 71, 72, 72, 73, 74	ofco 7429	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))))
100	99	eleq1d 2897	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
101	98, 100	sylibrd 261	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
102	101	expimpd 456	. . . . . 6 ⊢ (𝜑 → (((𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
103	26, 102	syl5bi 244	. . . . 5 ⊢ (𝜑 → (((𝑎 ∈ ran (1o eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1o eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
104	103	imp 409	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran (1o eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1o eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))) → ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
105		coeq1 5728	. . . . 5 ⊢ (𝑦 = ((𝐵 ↑m 1o) × {𝑎}) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
106	105	eleq1d 2897	. . . 4 ⊢ (𝑦 = ((𝐵 ↑m 1o) × {𝑎}) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
107		coeq1 5728	. . . . 5 ⊢ (𝑦 = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
108	107	eleq1d 2897	. . . 4 ⊢ (𝑦 = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
109		coeq1 5728	. . . . 5 ⊢ (𝑦 = 𝑎 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
110	109	eleq1d 2897	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
111		coeq1 5728	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
112	111	eleq1d 2897	. . . 4 ⊢ (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
113		coeq1 5728	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
114	113	eleq1d 2897	. . . 4 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
115		coeq1 5728	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
116	115	eleq1d 2897	. . . 4 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
117		coeq1 5728	. . . . 5 ⊢ (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
118	117	eleq1d 2897	. . . 4 ⊢ (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
119	16	pf1rcl 20512	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → 𝑅 ∈ CRing)
120	15, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
121	120	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ CRing)
122		1on 8109	. . . . . . . . . . . 12 ⊢ 1o ∈ On
123		eqid 2821	. . . . . . . . . . . . 13 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
124	123	mplassa 20235	. . . . . . . . . . . 12 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125	122, 120, 124	sylancr 589	. . . . . . . . . . 11 ⊢ (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126		eqid 2821	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
127		eqid 2821	. . . . . . . . . . . . 13 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
128	126, 127	ply1ascl 20426	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(1o mPoly 𝑅))
129		eqid 2821	. . . . . . . . . . . 12 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130	128, 129	asclrhm 20119	. . . . . . . . . . 11 ⊢ ((1o mPoly 𝑅) ∈ AssAlg → (algSc‘(Poly1‘𝑅)) ∈ ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
131	125, 130	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)) ∈ ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
132	122	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1o ∈ On)
133	123, 132, 120	mplsca 20225	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘(1o mPoly 𝑅)))
134	133	oveq1d 7171	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135	131, 134	eleqtrrd 2916	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
137	3, 136	rhmf 19478	. . . . . . . . 9 ⊢ ((algSc‘(Poly1‘𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)) → (algSc‘(Poly1‘𝑅)):𝐵⟶(Base‘(1o mPoly 𝑅)))
138	135, 137	syl 17	. . . . . . . 8 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)):𝐵⟶(Base‘(1o mPoly 𝑅)))
139	138	ffvelrnda 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140		eqid 2821	. . . . . . . 8 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
141	140, 23, 3, 123, 136	evl1val 20492	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ ((algSc‘(Poly1‘𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅))) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
142	121, 139, 141	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
143	140, 126, 3, 127	evl1sca 20497	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144	120, 143	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
145	3	ressid 16559	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
146	121, 145	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅)
147	146	oveq2d 7172	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅))
148	147	fveq2d 6674	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149	148, 128	syl6eqr 2874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(Poly1‘𝑅)))
150	149	fveq1d 6672	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎) = ((algSc‘(Poly1‘𝑅))‘𝑎))
151	150	fveq2d 6674	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)))
152		eqid 2821	. . . . . . . . 9 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵))
153		eqid 2821	. . . . . . . . 9 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
154		eqid 2821	. . . . . . . . 9 ⊢ (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly (𝑅 ↾s 𝐵)))
155	122	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 1o ∈ On)
156		crngring 19308	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
157	3	subrgid 19537	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158	120, 156, 157	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
159	158	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
161	24, 152, 153, 3, 154, 155, 121, 159, 160	evlssca 20302	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
162	151, 161	eqtr3d 2858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
163	162	coeq1d 5732	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
164	142, 144, 163	3eqtr3d 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
165		pf1ind.co	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
166		snex 5332	. . . . . . . . . 10 ⊢ {𝑓} ∈ V
167	4, 166	xpex 7476	. . . . . . . . 9 ⊢ (𝐵 × {𝑓}) ∈ V
168		pf1ind.wa	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
169	167, 168	elab 3667	. . . . . . . 8 ⊢ ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
170	165, 169	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
171	170	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
172		sneq 4577	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → {𝑓} = {𝑎})
173	172	xpeq2d 5585	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174	173	eleq1d 2897	. . . . . . 7 ⊢ (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓}))
175	174	rspccva 3622	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
176	171, 175	sylan 582	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
177	164, 176	eqeltrrd 2914	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
178		pf1ind.pr	. . . . . . . 8 ⊢ (𝜑 → 𝜃)
179		resiexg 7619	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
180	4, 179	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ 𝐵) ∈ V
181		pf1ind.wb	. . . . . . . . 9 ⊢ (𝑥 = ( I ↾ 𝐵) → (𝜓 ↔ 𝜃))
182	180, 181	elab 3667	. . . . . . . 8 ⊢ (( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
183	178, 182	sylibr 236	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓})
184	12, 183	eqeltrd 2913	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
185		el1o 8124	. . . . . . . . . 10 ⊢ (𝑎 ∈ 1o ↔ 𝑎 = ∅)
186		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑏‘𝑎) = (𝑏‘∅))
187	185, 186	sylbi 219	. . . . . . . . 9 ⊢ (𝑎 ∈ 1o → (𝑏‘𝑎) = (𝑏‘∅))
188	187	mpteq2dv 5162	. . . . . . . 8 ⊢ (𝑎 ∈ 1o → (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)))
189	188	coeq1d 5732	. . . . . . 7 ⊢ (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
190	189	eleq1d 2897	. . . . . 6 ⊢ (𝑎 ∈ 1o → (((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
191	184, 190	syl5ibrcom 249	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
192	191	imp 409	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
193	16, 3, 27	pf1mpf 20515	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∈ ran (1o eval 𝑅))
194	15, 193	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∈ ran (1o eval 𝑅))
195	3, 21, 22, 25, 80, 104, 106, 108, 110, 112, 114, 116, 118, 177, 192, 194	mpfind 20320	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
196	20, 195	eqeltrrd 2914	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
197		pf1ind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
198	197	elabg 3666	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
199	15, 198	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
200	196, 199	mpbid 234	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  Vcvv 3494  ∅c0 4291  {csn 4567   ↦ cmpt 5146   I cid 5459   × cxp 5553  ^◡ccnv 5554  ran crn 5556   ↾ cres 5557   ∘ ccom 5559  Oncon0 6191  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ∘_f cof 7407  1_oc1o 8095   ↑_m cmap 8406  Basecbs 16483   ↾_s cress 16484  +_gcplusg 16565  ._rcmulr 16566  Scalarcsca 16568  Ringcrg 19297  CRingccrg 19298   RingHom crh 19464  SubRingcsubrg 19531  AssAlgcasa 20082  algSccascl 20084   mPoly cmpl 20133   evalSub ces 20284   eval cevl 20285  Poly₁cpl1 20345  eval₁ce1 20477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-ofr 7410  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-sup 8906  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-0g 16715  df-gsum 16716  df-prds 16721  df-pws 16723  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-rnghom 19467  df-subrg 19533  df-lmod 19636  df-lss 19704  df-lsp 19744  df-assa 20085  df-asp 20086  df-ascl 20087  df-psr 20136  df-mvr 20137  df-mpl 20138  df-opsr 20140  df-evls 20286  df-evl 20287  df-psr1 20348  df-ply1 20350  df-evl1 20479

This theorem is referenced by:  pl1cn  31198