Theorem pf1ind 22398

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 6249	. . . . 5 ⊢ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
2		df1o2 8439	. . . . . . . . 9 ⊢ 1o = {∅}
3		pf1ind.cb	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
4	3	fvexi 6877	. . . . . . . . 9 ⊢ 𝐵 ∈ V
5		0ex 5256	. . . . . . . . 9 ⊢ ∅ ∈ V
6		eqid 2761	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))
7	2, 4, 5, 6	mapsncnv 8871	. . . . . . . 8 ⊢ ◡(𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
8	7	coeq2i 5830	. . . . . . 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 8872	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)):(𝐵 ↑m 1o)–1-1-onto→𝐵
10		f1ococnv2 6830	. . . . . . . 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 2810	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
13	12	coeq2d 5832	. . . . 5 ⊢ (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
14	1, 13	eqtrid 2808	. . . 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 22393	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → 𝐴:𝐵⟶𝐵)
18		fcoi1 6734	. . . . 5 ⊢ (𝐴:𝐵⟶𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
19	15, 17, 18	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
20	14, 19	eqtrd 2796	. . 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 2761	. . . . . 6 ⊢ (1o eval 𝑅) = (1o eval 𝑅)
24	23, 3	evlval 22133	. . . . 5 ⊢ (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
25	24	rneqi 5911	. . . 4 ⊢ ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26		an4 666	. . . . . 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 2761	. . . . . . . . . . . 12 ⊢ ran (1o eval 𝑅) = ran (1o eval 𝑅)
28	16, 3, 27	mpfpf1 22394	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
29	16, 3, 27	mpfpf1 22394	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30		vex 3457	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
31		pf1ind.wc	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
32	30, 31	elab 3638	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
33		eleq1 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
34	32, 33	bitr3id 287	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
35	34	anbi1d 640	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → ((𝜏 ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂)))
36	35	anbi1d 640	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (((𝜏 ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑)))
37		ovex 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f + 𝑔) ∈ V
38		pf1ind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f + 𝑔) → (𝜓 ↔ 𝜁))
39	37, 38	elab 3638	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
40		oveq1 7399	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
41	40	eleq1d 2846	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥 ∣ 𝜓}))
43	36, 42	imbi12d 346	. . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . . . 17 ⊢ 𝑔 ∈ V
45		pf1ind.wd	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
46	44, 45	elab 3638	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
47		eleq1 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
48	46, 47	bitr3id 287	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
49	48	anbi2d 639	. . . . . . . . . . . . . 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 640	. . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . 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 346	. . . . . . . . . . . 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 417	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜁))
56	55	an4s 670	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜁))
57	56	expimpd 457	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁))
58	43, 53, 57	vtocl2ga 3542	. . . . . . . . . . 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 605	. . . . . . . . . 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 420	. . . . . . . . 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 411	. . . . . . . 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 22145	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
63	62	ad2antrl 738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵 ↑m 1o)⟶𝐵)
64	63	ffnd 6688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵 ↑m 1o))
65	25, 3	mpff 22145	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
66	65	ad2antll 739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵 ↑m 1o)⟶𝐵)
67	66	ffnd 6688	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵 ↑m 1o))
68		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})) = (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))
69	2, 4, 5, 68	mapsnf1o3 8873	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤})):𝐵–1-1-onto→(𝐵 ↑m 1o)
70		f1of 6802	. . . . . . . . . . 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 7427	. . . . . . . . . 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 4178	. . . . . . . . . 10 ⊢ ((𝐵 ↑m 1o) ∩ (𝐵 ↑m 1o)) = (𝐵 ↑m 1o)
75	64, 67, 71, 72, 72, 73, 74	ofco 7681	. . . . . . . . 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 2846	. . . . . . . 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 457	. . . . . 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 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 410	. . . 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 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘f · 𝑔) ∈ V
82		pf1ind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘f · 𝑔) → (𝜓 ↔ 𝜎))
83	81, 82	elab 3638	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
84		oveq1 7399	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∘f · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
85	84	eleq1d 2846	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥 ∣ 𝜓}))
87	36, 86	imbi12d 346	. . . . . . . . . . . 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 7400	. . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . 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 346	. . . . . . . . . . . 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 417	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜎))
93	92	an4s 670	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜎))
94	93	expimpd 457	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎))
95	87, 90, 94	vtocl2ga 3542	. . . . . . . . . . 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 605	. . . . . . . . . 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 420	. . . . . . . . 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 411	. . . . . . . 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 7681	. . . . . . . . 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 2846	. . . . . . . 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 457	. . . . . 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 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 410	. . . 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 5827	. . . . 5 ⊢ (𝑦 = ((𝐵 ↑m 1o) × {𝑎}) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
106	105	eleq1d 2846	. . . 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 5827	. . . . 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 2846	. . . 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 5827	. . . . 5 ⊢ (𝑦 = 𝑎 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
110	109	eleq1d 2846	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
111		coeq1 5827	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
112	111	eleq1d 2846	. . . 4 ⊢ (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
113		coeq1 5827	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
114	113	eleq1d 2846	. . . 4 ⊢ (𝑦 = (𝑎 ∘f + 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
115		coeq1 5827	. . . . 5 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) = ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
116	115	eleq1d 2846	. . . 4 ⊢ (𝑦 = (𝑎 ∘f · 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘f · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
117		coeq1 5827	. . . . 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 2846	. . . 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 22392	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → 𝑅 ∈ CRing)
120	15, 119	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
121	120	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ CRing)
122		1on 8445	. . . . . . . . . . . 12 ⊢ 1o ∈ On
123		eqid 2761	. . . . . . . . . . . . 13 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
124	123	mplassa 22053	. . . . . . . . . . . 12 ⊢ ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125	122, 120, 124	sylancr 596	. . . . . . . . . . 11 ⊢ (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126		eqid 2761	. . . . . . . . . . . . 13 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
127		eqid 2761	. . . . . . . . . . . . 13 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(Poly1‘𝑅))
128	126, 127	ply1ascl 22301	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly1‘𝑅)) = (algSc‘(1o mPoly 𝑅))
129		eqid 2761	. . . . . . . . . . . 12 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130	128, 129	asclrhm 21922	. . . . . . . . . . 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 22044	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘(1o mPoly 𝑅)))
134	133	oveq1d 7407	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135	131, 134	eleqtrrd 2864	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(Poly1‘𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
137	3, 136	rhmf 20512	. . . . . . . . 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 7061	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(Poly1‘𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140		eqid 2761	. . . . . . . 8 ⊢ (eval1‘𝑅) = (eval1‘𝑅)
141	140, 23, 3, 123, 136	evl1val 22372	. . . . . . 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 593	. . . . . 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 22377	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144	120, 143	sylan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval1‘𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
145	3	ressid 17263	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
146	121, 145	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ↾s 𝐵) = 𝑅)
147	146	oveq2d 7408	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly 𝑅))
148	147	fveq2d 6867	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149	148, 128	eqtr4di 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1o mPoly (𝑅 ↾s 𝐵))) = (algSc‘(Poly1‘𝑅)))
150	149	fveq1d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎) = ((algSc‘(Poly1‘𝑅))‘𝑎))
151	150	fveq2d 6867	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)))
152		eqid 2761	. . . . . . . . 9 ⊢ (1o mPoly (𝑅 ↾s 𝐵)) = (1o mPoly (𝑅 ↾s 𝐵))
153		eqid 2761	. . . . . . . . 9 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
154		eqid 2761	. . . . . . . . 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 20274	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
157	3	subrgid 20602	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158	120, 156, 157	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
159	158	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
161	24, 152, 153, 3, 154, 155, 121, 159, 160	evlssca 22127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅 ↾s 𝐵)))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
162	151, 161	eqtr3d 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1‘𝑅))‘𝑎)) = ((𝐵 ↑m 1o) × {𝑎}))
163	162	coeq1d 5831	. . . . . 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 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) = (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))))
165		pf1ind.co	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
166		vsnex 5391	. . . . . . . . . 10 ⊢ {𝑓} ∈ V
167	4, 166	xpex 7732	. . . . . . . . 9 ⊢ (𝐵 × {𝑓}) ∈ V
168		pf1ind.wa	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
169	167, 168	elab 3638	. . . . . . . 8 ⊢ ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
170	165, 169	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
171	170	ralrimiva 3153	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
172		sneq 4591	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → {𝑓} = {𝑎})
173	172	xpeq2d 5675	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174	173	eleq1d 2846	. . . . . . 7 ⊢ (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓}))
175	174	rspccva 3580	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
176	171, 175	sylan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
177	164, 176	eqeltrrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝐵 ↑m 1o) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
178		pf1ind.pr	. . . . . . . 8 ⊢ (𝜑 → 𝜃)
179		resiexg 7889	. . . . . . . . . 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 3638	. . . . . . . 8 ⊢ (( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
183	178, 182	sylibr 236	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓})
184	12, 183	eqeltrd 2861	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
185		el1o 8459	. . . . . . . . . 10 ⊢ (𝑎 ∈ 1o ↔ 𝑎 = ∅)
186		fveq2 6863	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑏‘𝑎) = (𝑏‘∅))
187	185, 186	sylbi 219	. . . . . . . . 9 ⊢ (𝑎 ∈ 1o → (𝑏‘𝑎) = (𝑏‘∅))
188	187	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑎 ∈ 1o → (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) = (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅)))
189	188	coeq1d 5831	. . . . . . 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 2846	. . . . . 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 410	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
193	16, 3, 27	pf1mpf 22395	. . . . 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 22148	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑m 1o) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
196	20, 195	eqeltrrd 2862	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
197		pf1ind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
198	197	elabg 3635	. . 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 399   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  Vcvv 3453  ∅c0 4285  {csn 4581   ↦ cmpt 5180   I cid 5539   × cxp 5643  ^◡ccnv 5644  ran crn 5646   ↾ cres 5647   ∘ ccom 5649  Oncon0 6342  ⟶wf 6513  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392   ∘_f cof 7654  1_oc1o 8425   ↑_m cmap 8803  Basecbs 17228   ↾_s cress 17249  +_gcplusg 17269  ._rcmulr 17270  Scalarcsca 17272  Ringcrg 20262  CRingccrg 20263   RingHom crh 20497  SubRingcsubrg 20598  AssAlgcasa 21882  algSccascl 21884   mPoly cmpl 21938   evalSub ces 22105   eval cevl 22106  Poly₁cpl1 22219  eval₁ce1 22357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-ofr 7657  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-map 8805  df-pm 8806  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fsupp 9305  df-sup 9385  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-fzo 13657  df-seq 14012  df-hash 14341  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-hom 17293  df-cco 17294  df-0g 17453  df-gsum 17454  df-prds 17459  df-pws 17461  df-mre 17597  df-mrc 17598  df-acs 17600  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-submnd 18801  df-grp 18961  df-minusg 18962  df-sbg 18963  df-mulg 19093  df-subg 19148  df-ghm 19237  df-cntz 19340  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-srg 20216  df-ring 20264  df-cring 20265  df-rhm 20500  df-subrng 20575  df-subrg 20599  df-lmod 20909  df-lss 20979  df-lsp 21019  df-assa 21885  df-asp 21886  df-ascl 21887  df-psr 21941  df-mvr 21942  df-mpl 21943  df-opsr 21945  df-evls 22107  df-evl 22108  df-psr1 22222  df-ply1 22224  df-evl1 22359

This theorem is referenced by:  pl1cn  34213