Theorem pf1ind 22348

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 6224	. . . . 5 ⊢ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
2		df1o2 8409	. . . . . . . . 9 ⊢ 1o = {∅}
3		pf1ind.cb	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6848	. . . . . . . . 9 ⊢ 𝐵 ∈ V
5		0ex 5236	. . . . . . . . 9 ⊢ ∅ ∈ V
6		eqid 2740	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))
7	2, 4, 5, 6	mapsncnv 8838	. . . . . . . 8 ⊢ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
8	7	coeq2i 5809	. . . . . . 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 8839	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
10		f1ococnv2 6801	. . . . . . . 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	eqtr3id 2789	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
13	12	coeq2d 5811	. . . . 5 ⊢ (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
14	1, 13	eqtrid 2787	. . . 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 22343	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → 𝐴:𝐵⟶𝐵)
18		fcoi1 6708	. . . . 5 ⊢ (𝐴:𝐵⟶𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
19	15, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
20	14, 19	eqtrd 2775	. . 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 2740	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
24	23, 3	evlval 22083	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
25	24	rneqi 5886	. . . 4 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26		an4 662	. . . . . 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 2740	. . . . . . . . . . . 12 ⊢ ran (1o eval 𝑅) = ran (1o eval 𝑅)
28	16, 3, 27	mpfpf1 22344	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
29	16, 3, 27	mpfpf1 22344	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
31		pf1ind.wc	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
32	30, 31	elab 3624	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
33		eleq1 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
34	32, 33	bitr3id 286	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
35	34	anbi1d 637	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝜏 ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂)))
36	35	anbi1d 637	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝜏 ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑)))
37		ovex 7396	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
38		pf1ind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
39	37, 38	elab 3624	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
40		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
41	40	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
42	39, 41	bitr3id 286	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
43	36, 42	imbi12d 345	. . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑔 ∈ V
45		pf1ind.wd	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
46	44, 45	elab 3624	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
47		eleq1 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
48	46, 47	bitr3id 286	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
49	48	anbi2d 636	. . . . . . . . . . . . . 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 637	. . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . 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 345	. . . . . . . . . . . 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 414	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜁))
56	55	an4s 666	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜁))
57	56	expimpd 454	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁))
58	43, 53, 57	vtocl2ga 3524	. . . . . . . . . . 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 602	. . . . . . . . . 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 417	. . . . . . . . 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 408	. . . . . . . 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 22095	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
63	62	ad2antrl 734	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
64	63	ffnd 6663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵 ↑m 1o))
65	25, 3	mpff 22095	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
66	65	ad2antll 735	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
67	66	ffnd 6663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵 ↑m 1o))
68		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
69	2, 4, 5, 68	mapsnf1o3 8840	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵–1-1-onto→(𝐵 ↑m 1o)
70		f1of 6774	. . . . . . . . . . 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 7398	. . . . . . . . . 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 4162	. . . . . . . . . 10 ⊢ ((𝐵 ↑m 1o) ∩ (𝐵 ↑m 1o)) = (𝐵 ↑m 1o)
75	64, 67, 71, 72, 72, 73, 74	ofco 7652	. . . . . . . . 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 2825	. . . . . . . 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 260	. . . . . . 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 454	. . . . . 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	biimtrid 243	. . . . 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 407	. . . 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 7396	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
82		pf1ind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
83	81, 82	elab 3624	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
84		oveq1 7370	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
85	84	eleq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
86	83, 85	bitr3id 286	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
87	36, 86	imbi12d 345	. . . . . . . . . . . 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 7371	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . 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 345	. . . . . . . . . . . 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 414	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜎))
93	92	an4s 666	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜎))
94	93	expimpd 454	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎))
95	87, 90, 94	vtocl2ga 3524	. . . . . . . . . . 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 602	. . . . . . . . . 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 417	. . . . . . . . 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 408	. . . . . . . 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 7652	. . . . . . . . 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 2825	. . . . . . . 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 260	. . . . . . 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 454	. . . . . 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	biimtrid 243	. . . . 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 407	. . . 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 5806	. . . . 5 ⊢ (𝑦 = ((𝐵 ↑m 1o) × {𝑎}) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
106	105	eleq1d 2825	. . . 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 5806	. . . . 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 2825	. . . 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 5806	. . . . 5 ⊢ (𝑦 = 𝑎 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
110	109	eleq1d 2825	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
111		coeq1 5806	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
112	111	eleq1d 2825	. . . 4 ⊢ (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
113		coeq1 5806	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
114	113	eleq1d 2825	. . . 4 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
115		coeq1 5806	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
116	115	eleq1d 2825	. . . 4 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
117		coeq1 5806	. . . . 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 2825	. . . 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 22342	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → 𝑅 ∈ CRing)
120	15, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
121	120	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ CRing)
122		1on 8414	. . . . . . . . . . . 12 ⊢ 1o ∈ On
123		eqid 2740	. . . . . . . . . . . . 13 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
124	123	mplassa 22003	. . . . . . . . . . . 12 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125	122, 120, 124	sylancr 593	. . . . . . . . . . 11 ⊢ (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
127		eqid 2740	. . . . . . . . . . . . 13 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
128	126, 127	ply1ascl 22251	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(1o mPoly 𝑅))
129		eqid 2740	. . . . . . . . . . . 12 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130	128, 129	asclrhm 21872	. . . . . . . . . . 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 21994	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘(1o mPoly 𝑅)))
134	133	oveq1d 7378	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135	131, 134	eleqtrrd 2843	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
137	3, 136	rhmf 20462	. . . . . . . . 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	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140		eqid 2740	. . . . . . . 8 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
141	140, 23, 3, 123, 136	evl1val 22322	. . . . . . 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 590	. . . . . 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 22327	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144	120, 143	sylan 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
145	3	ressid 17212	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
146	121, 145	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅)
147	146	oveq2d 7379	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅))
148	147	fveq2d 6838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149	148, 128	eqtr4di 2793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(Poly1‘𝑅)))
150	149	fveq1d 6836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎) = ((algSc‘(Poly1‘𝑅))‘𝑎))
151	150	fveq2d 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)))
152		eqid 2740	. . . . . . . . 9 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵))
153		eqid 2740	. . . . . . . . 9 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
154		eqid 2740	. . . . . . . . 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 20224	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
157	3	subrgid 20552	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158	120, 156, 157	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
159	158	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
161	24, 152, 153, 3, 154, 155, 121, 159, 160	evlssca 22077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
162	151, 161	eqtr3d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
163	162	coeq1d 5810	. . . . . 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 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
165		pf1ind.co	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
166		vsnex 5371	. . . . . . . . . 10 ⊢ {𝑓} ∈ V
167	4, 166	xpex 7703	. . . . . . . . 9 ⊢ (𝐵 × {𝑓}) ∈ V
168		pf1ind.wa	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
169	167, 168	elab 3624	. . . . . . . 8 ⊢ ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
170	165, 169	sylibr 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
171	170	ralrimiva 3132	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
172		sneq 4572	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → {𝑓} = {𝑎})
173	172	xpeq2d 5655	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174	173	eleq1d 2825	. . . . . . 7 ⊢ (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓}))
175	174	rspccva 3566	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
176	171, 175	sylan 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
177	164, 176	eqeltrrd 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
178		pf1ind.pr	. . . . . . . 8 ⊢ (𝜑 → 𝜃)
179		resiexg 7859	. . . . . . . . . 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 3624	. . . . . . . 8 ⊢ (( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
183	178, 182	sylibr 235	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓})
184	12, 183	eqeltrd 2840	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
185		el1o 8427	. . . . . . . . . 10 ⊢ (𝑎 ∈ 1o ↔ 𝑎 = ∅)
186		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑏‘𝑎) = (𝑏‘∅))
187	185, 186	sylbi 218	. . . . . . . . 9 ⊢ (𝑎 ∈ 1o → (𝑏‘𝑎) = (𝑏‘∅))
188	187	mpteq2dv 5173	. . . . . . . 8 ⊢ (𝑎 ∈ 1o → (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)))
189	188	coeq1d 5810	. . . . . . 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 2825	. . . . . 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 248	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
192	191	imp 407	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
193	16, 3, 27	pf1mpf 22345	. . . . 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 22098	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
196	20, 195	eqeltrrd 2841	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
197		pf1ind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
198	197	elabg 3621	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
199	15, 198	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
200	196, 199	mpbid 233	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  Vcvv 3432  ∅c0 4268  {csn 4562   ↦ cmpt 5160   I cid 5519   × cxp 5623  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   ∘ ccom 5629  Oncon0 6317  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363   ∘_f cof 7625  1_oc1o 8395   ↑_m cmap 8770  Basecbs 17177   ↾_s cress 17198  +_gcplusg 17218  ._rcmulr 17219  Scalarcsca 17221  Ringcrg 20212  CRingccrg 20213   RingHom crh 20447  SubRingcsubrg 20548  AssAlgcasa 21832  algSccascl 21834   mPoly cmpl 21888   evalSub ces 22055   eval cevl 22056  Poly₁cpl1 22169  eval₁ce1 22307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-ofr 7628  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-pm 8773  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-sup 9352  df-oi 9422  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-z 12523  df-dec 12643  df-uz 12787  df-fz 13460  df-fzo 13607  df-seq 13962  df-hash 14291  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-hom 17242  df-cco 17243  df-0g 17402  df-gsum 17403  df-prds 17408  df-pws 17410  df-mre 17546  df-mrc 17547  df-acs 17549  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mhm 18749  df-submnd 18750  df-grp 18910  df-minusg 18911  df-sbg 18912  df-mulg 19042  df-subg 19097  df-ghm 19186  df-cntz 19290  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-srg 20166  df-ring 20214  df-cring 20215  df-rhm 20450  df-subrng 20525  df-subrg 20549  df-lmod 20859  df-lss 20929  df-lsp 20969  df-assa 21835  df-asp 21836  df-ascl 21837  df-psr 21891  df-mvr 21892  df-mpl 21893  df-opsr 21895  df-evls 22057  df-evl 22058  df-psr1 22172  df-ply1 22174  df-evl1 22309

This theorem is referenced by:  pl1cn  34146