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Theorem goalrlem 34387

Description: Lemma for goalr 34388 (induction step). (Contributed by AV, 22-Oct-2023.)

**Assertion**
Ref	Expression
goalrlem	⊢ (𝑁 ∈ ω → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))

Distinct variable groups: 𝑖,𝑁 𝑖,𝑎 Allowed substitution hint: 𝑁(𝑎)

Proof of Theorem goalrlem Dummy variables 𝑗 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7881	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2		df-goal 34333	. . . . . 6 ⊢ ∀_𝑔𝑖𝑎 = ⟨2_o, ⟨𝑖, 𝑎⟩⟩
3		opex 5465	. . . . . 6 ⊢ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ ∈ V
4	2, 3	eqeltri 2830	. . . . 5 ⊢ ∀_𝑔𝑖𝑎 ∈ V
5		isfmlasuc 34379	. . . . 5 ⊢ ((suc 𝑁 ∈ ω ∧ ∀_𝑔𝑖𝑎 ∈ V) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢))))
6	1, 4, 5	sylancl 587	. . . 4 ⊢ (𝑁 ∈ ω → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢))))
7	6	adantr 482	. . 3 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) ↔ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢))))
8		fmlasssuc 34380	. . . . . . . . . 10 ⊢ (suc 𝑁 ∈ ω → (Fmla‘suc 𝑁) ⊆ (Fmla‘suc suc 𝑁))
9	1, 8	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) ⊆ (Fmla‘suc suc 𝑁))
10	9	sseld 3982	. . . . . . . 8 ⊢ (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
11	10	com12 32	. . . . . . 7 ⊢ (𝑎 ∈ (Fmla‘suc 𝑁) → (𝑁 ∈ ω → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
12	11	imim2i 16	. . . . . 6 ⊢ ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → (𝑁 ∈ ω → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
13	12	com23 86	. . . . 5 ⊢ ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (𝑁 ∈ ω → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
14	13	impcom 409	. . . 4 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
15		gonanegoal 34343	. . . . . . . . . . 11 ⊢ (𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎
16		eqneqall 2952	. . . . . . . . . . 11 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → ((𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
17	15, 16	mpi 20	. . . . . . . . . 10 ⊢ ((𝑢⊼_𝑔𝑣) = ∀_𝑔𝑖𝑎 → 𝑎 ∈ (Fmla‘suc suc 𝑁))
18	17	eqcoms 2741	. . . . . . . . 9 ⊢ (∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc suc 𝑁))
19	18	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
20	19	rexlimdva 3156	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
21		df-goal 34333	. . . . . . . . . . . . 13 ⊢ ∀_𝑔𝑗𝑢 = ⟨2_o, ⟨𝑗, 𝑢⟩⟩
22	2, 21	eqeq12i 2751	. . . . . . . . . . . 12 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑗, 𝑢⟩⟩)
23		2oex 8477	. . . . . . . . . . . . 13 ⊢ 2_o ∈ V
24		opex 5465	. . . . . . . . . . . . 13 ⊢ ⟨𝑖, 𝑎⟩ ∈ V
25	23, 24	opth 5477	. . . . . . . . . . . 12 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑗, 𝑢⟩⟩ ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩))
26	22, 25	bitri 275	. . . . . . . . . . 11 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩))
27		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑖 ∈ V
28		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
29	27, 28	opth 5477	. . . . . . . . . . . 12 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩ ↔ (𝑖 = 𝑗 ∧ 𝑎 = 𝑢))
30		eleq1w 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
31	30	eqcoms 2741	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
32	31, 11	syl6bi 253	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘suc 𝑁) → (𝑁 ∈ ω → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
33	32	impcomd 413	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑢 → ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
34	29, 33	simplbiim 506	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩ → ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
35	26, 34	simplbiim 506	. . . . . . . . . 10 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
36	35	com12 32	. . . . . . . . 9 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
37	36	adantr 482	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
38	37	rexlimdva 3156	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
39	20, 38	jaod 858	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → ((∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
40	39	rexlimdva 3156	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
41	40	adantr 482	. . . 4 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → (∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
42	14, 41	jaod 858	. . 3 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) ∨ ∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
43	7, 42	sylbid 239	. 2 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
44	43	ex 414	1 ⊢ (𝑁 ∈ ω → ((∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 ⟨cop 4635 suc csuc 6367 ‘cfv 6544 (class class class)co 7409 ωcom 7855 2_oc2o 8460 ⊼_𝑔cgna 34325 ∀_𝑔cgol 34326 Fmlacfmla 34328

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-map 8822 df-goel 34331 df-gona 34332 df-goal 34333 df-sat 34334 df-fmla 34336

This theorem is referenced by: goalr 34388

W3C validator