Theorem pf1ind 22258

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 6218	. . . . 5 ⊢ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
2		df1o2 8402	. . . . . . . . 9 ⊢ 1o = {∅}
3		pf1ind.cb	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6840	. . . . . . . . 9 ⊢ 𝐵 ∈ V
5		0ex 5249	. . . . . . . . 9 ⊢ ∅ ∈ V
6		eqid 2729	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))
7	2, 4, 5, 6	mapsncnv 8827	. . . . . . . 8 ⊢ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
8	7	coeq2i 5807	. . . . . . 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 8828	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
10		f1ococnv2 6795	. . . . . . . 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 2778	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
13	12	coeq2d 5809	. . . . 5 ⊢ (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
14	1, 13	eqtrid 2776	. . . 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 22253	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → 𝐴:𝐵⟶𝐵)
18		fcoi1 6702	. . . . 5 ⊢ (𝐴:𝐵⟶𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
19	15, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
20	14, 19	eqtrd 2764	. . 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 2729	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
24	23, 3	evlval 22018	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
25	24	rneqi 5883	. . . 4 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26		an4 656	. . . . . 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 2729	. . . . . . . . . . . 12 ⊢ ran (1o eval 𝑅) = ran (1o eval 𝑅)
28	16, 3, 27	mpfpf1 22254	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
29	16, 3, 27	mpfpf1 22254	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30		vex 3442	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
31		pf1ind.wc	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
32	30, 31	elab 3637	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
33		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
34	32, 33	bitr3id 285	. . . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
38		pf1ind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
39	37, 38	elab 3637	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
40		oveq1 7360	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
41	40	eleq1d 2813	. . . . . . . . . . . . . 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 285	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
43	36, 42	imbi12d 344	. . . . . . . . . . . 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 3442	. . . . . . . . . . . . . . . . 17 ⊢ 𝑔 ∈ V
45		pf1ind.wd	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
46	44, 45	elab 3637	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
47		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
48	46, 47	bitr3id 285	. . . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 413	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜁))
56	55	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜁))
57	56	expimpd 453	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁))
58	43, 53, 57	vtocl2ga 3535	. . . . . . . . . . 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 596	. . . . . . . . . 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 416	. . . . . . . . 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 407	. . . . . . . 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 22027	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
63	62	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
64	63	ffnd 6657	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵 ↑m 1o))
65	25, 3	mpff 22027	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
66	65	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
67	66	ffnd 6657	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵 ↑m 1o))
68		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
69	2, 4, 5, 68	mapsnf1o3 8829	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵–1-1-onto→(𝐵 ↑m 1o)
70		f1of 6768	. . . . . . . . . . 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 7388	. . . . . . . . . 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 4180	. . . . . . . . . 10 ⊢ ((𝐵 ↑m 1o) ∩ (𝐵 ↑m 1o)) = (𝐵 ↑m 1o)
75	64, 67, 71, 72, 72, 73, 74	ofco 7642	. . . . . . . . 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 2813	. . . . . . . 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 259	. . . . . . 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 453	. . . . . 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 242	. . . . 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 406	. . . 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 7386	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
82		pf1ind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
83	81, 82	elab 3637	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
84		oveq1 7360	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
85	84	eleq1d 2813	. . . . . . . . . . . . . 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 285	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
87	36, 86	imbi12d 344	. . . . . . . . . . . 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 7361	. . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 413	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜎))
93	92	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜎))
94	93	expimpd 453	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎))
95	87, 90, 94	vtocl2ga 3535	. . . . . . . . . . 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 596	. . . . . . . . . 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 416	. . . . . . . . 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 407	. . . . . . . 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 7642	. . . . . . . . 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 2813	. . . . . . . 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 259	. . . . . . 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 453	. . . . . 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 242	. . . . 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 406	. . . 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 5804	. . . . 5 ⊢ (𝑦 = ((𝐵 ↑m 1o) × {𝑎}) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
106	105	eleq1d 2813	. . . 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 5804	. . . . 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 2813	. . . 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 5804	. . . . 5 ⊢ (𝑦 = 𝑎 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
110	109	eleq1d 2813	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
111		coeq1 5804	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
112	111	eleq1d 2813	. . . 4 ⊢ (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
113		coeq1 5804	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
114	113	eleq1d 2813	. . . 4 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
115		coeq1 5804	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
116	115	eleq1d 2813	. . . 4 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
117		coeq1 5804	. . . . 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 2813	. . . 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 22252	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → 𝑅 ∈ CRing)
120	15, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
121	120	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ CRing)
122		1on 8407	. . . . . . . . . . . 12 ⊢ 1o ∈ On
123		eqid 2729	. . . . . . . . . . . . 13 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
124	123	mplassa 21947	. . . . . . . . . . . 12 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125	122, 120, 124	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
127		eqid 2729	. . . . . . . . . . . . 13 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
128	126, 127	ply1ascl 22160	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(1o mPoly 𝑅))
129		eqid 2729	. . . . . . . . . . . 12 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130	128, 129	asclrhm 21815	. . . . . . . . . . 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 21938	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘(1o mPoly 𝑅)))
134	133	oveq1d 7368	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135	131, 134	eleqtrrd 2831	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
137	3, 136	rhmf 20388	. . . . . . . . 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 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140		eqid 2729	. . . . . . . 8 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
141	140, 23, 3, 123, 136	evl1val 22232	. . . . . . 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 584	. . . . . 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 22237	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144	120, 143	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
145	3	ressid 17173	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
146	121, 145	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅)
147	146	oveq2d 7369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅))
148	147	fveq2d 6830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149	148, 128	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(Poly1‘𝑅)))
150	149	fveq1d 6828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎) = ((algSc‘(Poly1‘𝑅))‘𝑎))
151	150	fveq2d 6830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)))
152		eqid 2729	. . . . . . . . 9 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵))
153		eqid 2729	. . . . . . . . 9 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
154		eqid 2729	. . . . . . . . 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 20148	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
157	3	subrgid 20476	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158	120, 156, 157	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
159	158	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
161	24, 152, 153, 3, 154, 155, 121, 159, 160	evlssca 22012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
162	151, 161	eqtr3d 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
163	162	coeq1d 5808	. . . . . 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 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
165		pf1ind.co	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
166		vsnex 5376	. . . . . . . . . 10 ⊢ {𝑓} ∈ V
167	4, 166	xpex 7693	. . . . . . . . 9 ⊢ (𝐵 × {𝑓}) ∈ V
168		pf1ind.wa	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
169	167, 168	elab 3637	. . . . . . . 8 ⊢ ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
170	165, 169	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
171	170	ralrimiva 3121	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
172		sneq 4589	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → {𝑓} = {𝑎})
173	172	xpeq2d 5653	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174	173	eleq1d 2813	. . . . . . 7 ⊢ (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓}))
175	174	rspccva 3578	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
176	171, 175	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
177	164, 176	eqeltrrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
178		pf1ind.pr	. . . . . . . 8 ⊢ (𝜑 → 𝜃)
179		resiexg 7852	. . . . . . . . . 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 3637	. . . . . . . 8 ⊢ (( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
183	178, 182	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓})
184	12, 183	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
185		el1o 8420	. . . . . . . . . 10 ⊢ (𝑎 ∈ 1o ↔ 𝑎 = ∅)
186		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑏‘𝑎) = (𝑏‘∅))
187	185, 186	sylbi 217	. . . . . . . . 9 ⊢ (𝑎 ∈ 1o → (𝑏‘𝑎) = (𝑏‘∅))
188	187	mpteq2dv 5189	. . . . . . . 8 ⊢ (𝑎 ∈ 1o → (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)))
189	188	coeq1d 5808	. . . . . . 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 2813	. . . . . 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 247	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
192	191	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
193	16, 3, 27	pf1mpf 22255	. . . . 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 22030	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
196	20, 195	eqeltrrd 2829	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
197		pf1ind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
198	197	elabg 3634	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
199	15, 198	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
200	196, 199	mpbid 232	1 ⊢ (𝜑 → 𝜌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3438  ∅c0 4286  {csn 4579   ↦ cmpt 5176   I cid 5517   × cxp 5621  ^◡ccnv 5622  ran crn 5624   ↾ cres 5625   ∘ ccom 5627  Oncon0 6311  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353   ∘_f cof 7615  1_oc1o 8388   ↑_m cmap 8760  Basecbs 17138   ↾_s cress 17159  +_gcplusg 17179  ._rcmulr 17180  Scalarcsca 17182  Ringcrg 20136  CRingccrg 20137   RingHom crh 20372  SubRingcsubrg 20472  AssAlgcasa 21775  algSccascl 21777   mPoly cmpl 21831   evalSub ces 21995   eval cevl 21996  Poly₁cpl1 22077  eval₁ce1 22217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617  df-ofr 7618  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-map 8762  df-pm 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9271  df-sup 9351  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-z 12490  df-dec 12610  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-hom 17203  df-cco 17204  df-0g 17363  df-gsum 17364  df-prds 17369  df-pws 17371  df-mre 17506  df-mrc 17507  df-acs 17509  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-mhm 18675  df-submnd 18676  df-grp 18833  df-minusg 18834  df-sbg 18835  df-mulg 18965  df-subg 19020  df-ghm 19110  df-cntz 19214  df-cmn 19679  df-abl 19680  df-mgp 20044  df-rng 20056  df-ur 20085  df-srg 20090  df-ring 20138  df-cring 20139  df-rhm 20375  df-subrng 20449  df-subrg 20473  df-lmod 20783  df-lss 20853  df-lsp 20893  df-assa 21778  df-asp 21779  df-ascl 21780  df-psr 21834  df-mvr 21835  df-mpl 21836  df-opsr 21838  df-evls 21997  df-evl 21998  df-psr1 22080  df-ply1 22082  df-evl1 22219

This theorem is referenced by:  pl1cn  33921