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Theorem pf1ind 19540

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 5571	. . . . 5 ⊢ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
2		df1o2 7459	. . . . . . . . 9 ⊢ 1_𝑜 = {∅}
3		pf1ind.cb	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
4		fvex 6113	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
5	3, 4	eqeltri 2684	. . . . . . . . 9 ⊢ 𝐵 ∈ V
6		0ex 4718	. . . . . . . . 9 ⊢ ∅ ∈ V
7		eqid 2610	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))
8	2, 5, 6, 7	mapsncnv 7790	. . . . . . . 8 ⊢ ^◡(𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) = (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))
9	8	coeq2i 5204	. . . . . . 7 ⊢ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ ^◡(𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))
10	2, 5, 6, 7	mapsnf1o2 7791	. . . . . . . 8 ⊢ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)):(𝐵 ↑_𝑚 1_𝑜)–1-1-onto→𝐵
11		f1ococnv2 6076	. . . . . . . 8 ⊢ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)):(𝐵 ↑_𝑚 1_𝑜)–1-1-onto→𝐵 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ ^◡(𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
12	10, 11	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ ^◡(𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
13	9, 12	syl5eqr 2658	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ( I ↾ 𝐵))
14	13	coeq2d 5206	. . . . 5 ⊢ (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
15	1, 14	syl5eq 2656	. . . 4 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (𝐴 ∘ ( I ↾ 𝐵)))
16		pf1ind.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑄)
17		pf1ind.cq	. . . . . 6 ⊢ 𝑄 = ran (eval₁‘𝑅)
18	17, 3	pf1f 19535	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → 𝐴:𝐵⟶𝐵)
19		fcoi1 5991	. . . . 5 ⊢ (𝐴:𝐵⟶𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
20	16, 18, 19	3syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
21	15, 20	eqtrd 2644	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = 𝐴)
22		pf1ind.cp	. . . 4 ⊢ + = (+_g‘𝑅)
23		pf1ind.ct	. . . 4 ⊢ · = (._r‘𝑅)
24		eqid 2610	. . . . . 6 ⊢ (1_𝑜 eval 𝑅) = (1_𝑜 eval 𝑅)
25	24, 3	evlval 19345	. . . . 5 ⊢ (1_𝑜 eval 𝑅) = ((1_𝑜 evalSub 𝑅)‘𝐵)
26	25	rneqi 5273	. . . 4 ⊢ ran (1_𝑜 eval 𝑅) = ran ((1_𝑜 evalSub 𝑅)‘𝐵)
27		an4 861	. . . . . 6 ⊢ (((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) ↔ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})))
28		eqid 2610	. . . . . . . . . . . 12 ⊢ ran (1_𝑜 eval 𝑅) = ran (1_𝑜 eval 𝑅)
29	17, 3, 28	mpfpf1 19536	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ran (1_𝑜 eval 𝑅) → (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄)
30	17, 3, 28	mpfpf1 19536	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran (1_𝑜 eval 𝑅) → (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄)
31		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
32		pf1ind.wc	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑓 → (𝜓 ↔ 𝜏))
33	31, 32	elab 3319	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ 𝜏)
34		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝑓 ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
35	33, 34	syl5bbr 273	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
36	35	anbi1d 737	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((𝜏 ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂)))
37	36	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((𝜏 ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑)))
38		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘_𝑓 + 𝑔) ∈ V
39		pf1ind.we	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘_𝑓 + 𝑔) → (𝜓 ↔ 𝜁))
40	38, 39	elab 3319	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜁)
41		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝑓 ∘_𝑓 + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔))
42	41	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((𝑓 ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓}))
43	40, 42	syl5bbr 273	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓}))
44	37, 43	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓})))
45		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑔 ∈ V
46		pf1ind.wd	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑔 → (𝜓 ↔ 𝜂))
47	45, 46	elab 3319	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ 𝜂)
48		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝑔 ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
49	47, 48	syl5bbr 273	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
50	49	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})))
51	50	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑)))
52		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))))
53	52	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
54	51, 53	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + 𝑔) ∈ {𝑥 ∣ 𝜓}) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
55		pf1ind.ad	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜁)
56	55	expcom 450	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜁))
57	56	an4s 865	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜁))
58	57	expimpd 627	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜁))
59	44, 54, 58	vtocl2ga 3247	. . . . . . . . . . 11 ⊢ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
60	29, 30, 59	syl2an 493	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
61	60	expcomd 453	. . . . . . . . 9 ⊢ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
62	61	impcom 445	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
63	26, 3	mpff 19354	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran (1_𝑜 eval 𝑅) → 𝑎:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵)
64	63	ad2antrl 760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → 𝑎:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵)
65		ffn 5958	. . . . . . . . . . 11 ⊢ (𝑎:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵 → 𝑎 Fn (𝐵 ↑_𝑚 1_𝑜))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → 𝑎 Fn (𝐵 ↑_𝑚 1_𝑜))
67	26, 3	mpff 19354	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ran (1_𝑜 eval 𝑅) → 𝑏:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵)
68	67	ad2antll 761	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → 𝑏:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵)
69		ffn 5958	. . . . . . . . . . 11 ⊢ (𝑏:(𝐵 ↑_𝑚 1_𝑜)⟶𝐵 → 𝑏 Fn (𝐵 ↑_𝑚 1_𝑜))
70	68, 69	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → 𝑏 Fn (𝐵 ↑_𝑚 1_𝑜))
71		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})) = (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))
72	2, 5, 6, 71	mapsnf1o3 7792	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})):𝐵–1-1-onto→(𝐵 ↑_𝑚 1_𝑜)
73		f1of 6050	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})):𝐵–1-1-onto→(𝐵 ↑_𝑚 1_𝑜) → (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})):𝐵⟶(𝐵 ↑_𝑚 1_𝑜))
74	72, 73	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})):𝐵⟶(𝐵 ↑_𝑚 1_𝑜))
75		ovex 6577	. . . . . . . . . . 11 ⊢ (𝐵 ↑_𝑚 1_𝑜) ∈ V
76	75	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (𝐵 ↑_𝑚 1_𝑜) ∈ V)
77	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → 𝐵 ∈ V)
78		inidm 3784	. . . . . . . . . 10 ⊢ ((𝐵 ↑_𝑚 1_𝑜) ∩ (𝐵 ↑_𝑚 1_𝑜)) = (𝐵 ↑_𝑚 1_𝑜)
79	66, 70, 74, 76, 76, 77, 78	ofco 6815	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))))
80	79	eleq1d 2672	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 + (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
81	62, 80	sylibrd 248	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
82	81	expimpd 627	. . . . . 6 ⊢ (𝜑 → (((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
83	27, 82	syl5bi 231	. . . . 5 ⊢ (𝜑 → (((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
84	83	imp 444	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))) → ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
85		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∘_𝑓 · 𝑔) ∈ V
86		pf1ind.wf	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑓 ∘_𝑓 · 𝑔) → (𝜓 ↔ 𝜎))
87	85, 86	elab 3319	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ 𝜎)
88		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝑓 ∘_𝑓 · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔))
89	88	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((𝑓 ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓}))
90	87, 89	syl5bbr 273	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓}))
91	37, 90	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓})))
92		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))))
93	92	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
94	51, 93	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) → (((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · 𝑔) ∈ {𝑥 ∣ 𝜓}) ↔ ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
95		pf1ind.mu	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂))) → 𝜎)
96	95	expcom 450	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝜏) ∧ (𝑔 ∈ 𝑄 ∧ 𝜂)) → (𝜑 → 𝜎))
97	96	an4s 865	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) ∧ (𝜏 ∧ 𝜂)) → (𝜑 → 𝜎))
98	97	expimpd 627	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ 𝑄 ∧ 𝑔 ∈ 𝑄) → (((𝜏 ∧ 𝜂) ∧ 𝜑) → 𝜎))
99	91, 94, 98	vtocl2ga 3247	. . . . . . . . . . 11 ⊢ (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
100	29, 30, 99	syl2an 493	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) → ((((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
101	100	expcomd 453	. . . . . . . . 9 ⊢ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓})))
102	101	impcom 445	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
103	66, 70, 74, 76, 76, 77, 78	ofco 6815	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))))
104	103	eleq1d 2672	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∘_𝑓 · (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤})))) ∈ {𝑥 ∣ 𝜓}))
105	102, 104	sylibrd 248	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) → ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
106	105	expimpd 627	. . . . . 6 ⊢ (𝜑 → (((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1_𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
107	27, 106	syl5bi 231	. . . . 5 ⊢ (𝜑 → (((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})) → ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
108	107	imp 444	. . . 4 ⊢ ((𝜑 ∧ ((𝑎 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}) ∧ (𝑏 ∈ ran (1_𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))) → ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
109		coeq1 5201	. . . . 5 ⊢ (𝑦 = ((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
110	109	eleq1d 2672	. . . 4 ⊢ (𝑦 = ((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
111		coeq1 5201	. . . . 5 ⊢ (𝑦 = (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
112	111	eleq1d 2672	. . . 4 ⊢ (𝑦 = (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
113		coeq1 5201	. . . . 5 ⊢ (𝑦 = 𝑎 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
114	113	eleq1d 2672	. . . 4 ⊢ (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑎 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
115		coeq1 5201	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
116	115	eleq1d 2672	. . . 4 ⊢ (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ (𝑏 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
117		coeq1 5201	. . . . 5 ⊢ (𝑦 = (𝑎 ∘_𝑓 + 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
118	117	eleq1d 2672	. . . 4 ⊢ (𝑦 = (𝑎 ∘_𝑓 + 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘_𝑓 + 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
119		coeq1 5201	. . . . 5 ⊢ (𝑦 = (𝑎 ∘_𝑓 · 𝑏) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
120	119	eleq1d 2672	. . . 4 ⊢ (𝑦 = (𝑎 ∘_𝑓 · 𝑏) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑎 ∘_𝑓 · 𝑏) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
121		coeq1 5201	. . . . 5 ⊢ (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
122	121	eleq1d 2672	. . . 4 ⊢ (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
123	17	pf1rcl 19534	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑄 → 𝑅 ∈ CRing)
124	16, 123	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing)
125	124	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑅 ∈ CRing)
126		1on 7454	. . . . . . . . . . . 12 ⊢ 1_𝑜 ∈ On
127		eqid 2610	. . . . . . . . . . . . 13 ⊢ (1_𝑜 mPoly 𝑅) = (1_𝑜 mPoly 𝑅)
128	127	mplassa 19275	. . . . . . . . . . . 12 ⊢ ((1_𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1_𝑜 mPoly 𝑅) ∈ AssAlg)
129	126, 124, 128	sylancr 694	. . . . . . . . . . 11 ⊢ (𝜑 → (1_𝑜 mPoly 𝑅) ∈ AssAlg)
130		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
131		eqid 2610	. . . . . . . . . . . . 13 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
132	130, 131	ply1ascl 19449	. . . . . . . . . . . 12 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(1_𝑜 mPoly 𝑅))
133		eqid 2610	. . . . . . . . . . . 12 ⊢ (Scalar‘(1_𝑜 mPoly 𝑅)) = (Scalar‘(1_𝑜 mPoly 𝑅))
134	132, 133	asclrhm 19163	. . . . . . . . . . 11 ⊢ ((1_𝑜 mPoly 𝑅) ∈ AssAlg → (algSc‘(Poly₁‘𝑅)) ∈ ((Scalar‘(1_𝑜 mPoly 𝑅)) RingHom (1_𝑜 mPoly 𝑅)))
135	129, 134	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (algSc‘(Poly₁‘𝑅)) ∈ ((Scalar‘(1_𝑜 mPoly 𝑅)) RingHom (1_𝑜 mPoly 𝑅)))
136	126	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1_𝑜 ∈ On)
137	127, 136, 124	mplsca 19266	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘(1_𝑜 mPoly 𝑅)))
138	137	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 RingHom (1_𝑜 mPoly 𝑅)) = ((Scalar‘(1_𝑜 mPoly 𝑅)) RingHom (1_𝑜 mPoly 𝑅)))
139	135, 138	eleqtrrd 2691	. . . . . . . . 9 ⊢ (𝜑 → (algSc‘(Poly₁‘𝑅)) ∈ (𝑅 RingHom (1_𝑜 mPoly 𝑅)))
140		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘(1_𝑜 mPoly 𝑅)) = (Base‘(1_𝑜 mPoly 𝑅))
141	3, 140	rhmf 18549	. . . . . . . . 9 ⊢ ((algSc‘(Poly₁‘𝑅)) ∈ (𝑅 RingHom (1_𝑜 mPoly 𝑅)) → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(1_𝑜 mPoly 𝑅)))
142	139, 141	syl 17	. . . . . . . 8 ⊢ (𝜑 → (algSc‘(Poly₁‘𝑅)):𝐵⟶(Base‘(1_𝑜 mPoly 𝑅)))
143	142	ffvelrnda 6267	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(Poly₁‘𝑅))‘𝑎) ∈ (Base‘(1_𝑜 mPoly 𝑅)))
144		eqid 2610	. . . . . . . 8 ⊢ (eval₁‘𝑅) = (eval₁‘𝑅)
145	144, 24, 3, 127, 140	evl1val 19514	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ ((algSc‘(Poly₁‘𝑅))‘𝑎) ∈ (Base‘(1_𝑜 mPoly 𝑅))) → ((eval₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) = (((1_𝑜 eval 𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
146	125, 143, 145	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) = (((1_𝑜 eval 𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
147	144, 130, 3, 131	evl1sca 19519	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝐵) → ((eval₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
148	124, 147	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((eval₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) = (𝐵 × {𝑎}))
149	3	ressid 15762	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑅 ↾_s 𝐵) = 𝑅)
150	125, 149	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑅 ↾_s 𝐵) = 𝑅)
151	150	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (1_𝑜 mPoly (𝑅 ↾_s 𝐵)) = (1_𝑜 mPoly 𝑅))
152	151	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_𝑜 mPoly 𝑅)))
153	152, 132	syl6eqr 2662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(Poly₁‘𝑅)))
154	153	fveq1d 6105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵)))‘𝑎) = ((algSc‘(Poly₁‘𝑅))‘𝑎))
155	154	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1_𝑜 eval 𝑅)‘((algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵)))‘𝑎)) = ((1_𝑜 eval 𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)))
156		eqid 2610	. . . . . . . . 9 ⊢ (1_𝑜 mPoly (𝑅 ↾_s 𝐵)) = (1_𝑜 mPoly (𝑅 ↾_s 𝐵))
157		eqid 2610	. . . . . . . . 9 ⊢ (𝑅 ↾_s 𝐵) = (𝑅 ↾_s 𝐵)
158		eqid 2610	. . . . . . . . 9 ⊢ (algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵))) = (algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵)))
159	126	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 1_𝑜 ∈ On)
160		crngring 18381	. . . . . . . . . . 11 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
161	3	subrgid 18605	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
162	124, 160, 161	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
163	162	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ (SubRing‘𝑅))
164		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵)
165	25, 156, 157, 3, 158, 159, 125, 163, 164	evlssca 19343	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1_𝑜 eval 𝑅)‘((algSc‘(1_𝑜 mPoly (𝑅 ↾_s 𝐵)))‘𝑎)) = ((𝐵 ↑_𝑚 1_𝑜) × {𝑎}))
166	155, 165	eqtr3d 2646	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((1_𝑜 eval 𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) = ((𝐵 ↑_𝑚 1_𝑜) × {𝑎}))
167	166	coeq1d 5205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((1_𝑜 eval 𝑅)‘((algSc‘(Poly₁‘𝑅))‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = (((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
168	146, 148, 167	3eqtr3d 2652	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) = (((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
169		pf1ind.co	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → 𝜒)
170		snex 4835	. . . . . . . . . 10 ⊢ {𝑓} ∈ V
171	5, 170	xpex 6860	. . . . . . . . 9 ⊢ (𝐵 × {𝑓}) ∈ V
172		pf1ind.wa	. . . . . . . . 9 ⊢ (𝑥 = (𝐵 × {𝑓}) → (𝜓 ↔ 𝜒))
173	171, 172	elab 3319	. . . . . . . 8 ⊢ ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)
174	169, 173	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐵) → (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
175	174	ralrimiva 2949	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓})
176		sneq 4135	. . . . . . . . 9 ⊢ (𝑓 = 𝑎 → {𝑓} = {𝑎})
177	176	xpeq2d 5063	. . . . . . . 8 ⊢ (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
178	177	eleq1d 2672	. . . . . . 7 ⊢ (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓}))
179	178	rspccva 3281	. . . . . 6 ⊢ ((∀𝑓 ∈ 𝐵 (𝐵 × {𝑓}) ∈ {𝑥 ∣ 𝜓} ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
180	175, 179	sylan 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐵 × {𝑎}) ∈ {𝑥 ∣ 𝜓})
181	168, 180	eqeltrrd 2689	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝐵 ↑_𝑚 1_𝑜) × {𝑎}) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
182		pf1ind.pr	. . . . . . . 8 ⊢ (𝜑 → 𝜃)
183		resiexg 6994	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
184	5, 183	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ 𝐵) ∈ V
185		pf1ind.wb	. . . . . . . . 9 ⊢ (𝑥 = ( I ↾ 𝐵) → (𝜓 ↔ 𝜃))
186	184, 185	elab 3319	. . . . . . . 8 ⊢ (( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓} ↔ 𝜃)
187	182, 186	sylibr 223	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐵) ∈ {𝑥 ∣ 𝜓})
188	13, 187	eqeltrd 2688	. . . . . 6 ⊢ (𝜑 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
189		el1o 7466	. . . . . . . . . 10 ⊢ (𝑎 ∈ 1_𝑜 ↔ 𝑎 = ∅)
190		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑏‘𝑎) = (𝑏‘∅))
191	189, 190	sylbi 206	. . . . . . . . 9 ⊢ (𝑎 ∈ 1_𝑜 → (𝑏‘𝑎) = (𝑏‘∅))
192	191	mpteq2dv 4673	. . . . . . . 8 ⊢ (𝑎 ∈ 1_𝑜 → (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) = (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)))
193	192	coeq1d 5205	. . . . . . 7 ⊢ (𝑎 ∈ 1_𝑜 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))))
194	193	eleq1d 2672	. . . . . 6 ⊢ (𝑎 ∈ 1_𝑜 → (((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓} ↔ ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
195	188, 194	syl5ibrcom 236	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 1_𝑜 → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓}))
196	195	imp 444	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 1_𝑜) → ((𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘𝑎)) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
197	17, 3, 28	pf1mpf 19537	. . . . 5 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∈ ran (1_𝑜 eval 𝑅))
198	16, 197	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∈ ran (1_𝑜 eval 𝑅))
199	3, 22, 23, 26, 84, 108, 110, 112, 114, 116, 118, 120, 122, 181, 196, 198	mpfind 19357	. . 3 ⊢ (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵 ↑_𝑚 1_𝑜) ↦ (𝑏‘∅))) ∘ (𝑤 ∈ 𝐵 ↦ (1_𝑜 × {𝑤}))) ∈ {𝑥 ∣ 𝜓})
200	21, 199	eqeltrrd 2689	. 2 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
201		pf1ind.wg	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜌))
202	201	elabg 3320	. . 3 ⊢ (𝐴 ∈ 𝑄 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
203	16, 202	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜌))
204	200, 203	mpbid 221	1 ⊢ (𝜑 → 𝜌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 Vcvv 3173 ∅c0 3874 {csn 4125 ↦ cmpt 4643 I cid 4948 × cxp 5036 ^◡ccnv 5037 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Oncon0 5640 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 1_𝑜c1o 7440 ↑_𝑚 cmap 7744 Basecbs 15695 ↾_s cress 15696 +_gcplusg 15768 ._rcmulr 15769 Scalarcsca 15771 Ringcrg 18370 CRingccrg 18371 RingHom crh 18535 SubRingcsubrg 18599 AssAlgcasa 19130 algSccascl 19132 mPoly cmpl 19174 evalSub ces 19325 eval cevl 19326 Poly₁cpl1 19368 eval₁ce1 19500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-srg 18329 df-ring 18372 df-cring 18373 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-assa 19133 df-asp 19134 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-evls 19327 df-evl 19328 df-psr1 19371 df-ply1 19373 df-evl1 19502

This theorem is referenced by: pl1cn 29329

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