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Theorem goalrlem 34685

Description: Lemma for goalr 34686 (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 7883	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2		df-goal 34631	. . . . . 6 ⊢ ∀_𝑔𝑖𝑎 = ⟨2_o, ⟨𝑖, 𝑎⟩⟩
3		opex 5463	. . . . . 6 ⊢ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ ∈ V
4	2, 3	eqeltri 2827	. . . . 5 ⊢ ∀_𝑔𝑖𝑎 ∈ V
5		isfmlasuc 34677	. . . . 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 584	. . . 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 479	. . 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 34678	. . . . . . . . . 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 3980	. . . . . . . 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 406	. . . 4 ⊢ ((𝑁 ∈ ω ∧ (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁))) → (∀_𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
15		gonanegoal 34641	. . . . . . . . . . 11 ⊢ (𝑢⊼_𝑔𝑣) ≠ ∀_𝑔𝑖𝑎
16		eqneqall 2949	. . . . . . . . . . 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 2738	. . . . . . . . 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 3153	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑣 ∈ (Fmla‘suc 𝑁)∀_𝑔𝑖𝑎 = (𝑢⊼_𝑔𝑣) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
21		df-goal 34631	. . . . . . . . . . . . 13 ⊢ ∀_𝑔𝑗𝑢 = ⟨2_o, ⟨𝑗, 𝑢⟩⟩
22	2, 21	eqeq12i 2748	. . . . . . . . . . . 12 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 ↔ ⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑗, 𝑢⟩⟩)
23		2oex 8479	. . . . . . . . . . . . 13 ⊢ 2_o ∈ V
24		opex 5463	. . . . . . . . . . . . 13 ⊢ ⟨𝑖, 𝑎⟩ ∈ V
25	23, 24	opth 5475	. . . . . . . . . . . 12 ⊢ (⟨2_o, ⟨𝑖, 𝑎⟩⟩ = ⟨2_o, ⟨𝑗, 𝑢⟩⟩ ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩))
26	22, 25	bitri 274	. . . . . . . . . . 11 ⊢ (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 ↔ (2_o = 2_o ∧ ⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩))
27		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑖 ∈ V
28		vex 3476	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
29	27, 28	opth 5475	. . . . . . . . . . . 12 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩ ↔ (𝑖 = 𝑗 ∧ 𝑎 = 𝑢))
30		eleq1w 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
31	30	eqcoms 2738	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
32	31, 11	syl6bi 252	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘suc 𝑁) → (𝑁 ∈ ω → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
33	32	impcomd 410	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑢 → ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
34	29, 33	simplbiim 503	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑎⟩ = ⟨𝑗, 𝑢⟩ → ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
35	26, 34	simplbiim 503	. . . . . . . . . 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 479	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑗 ∈ ω) → (∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
38	37	rexlimdva 3153	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑗 ∈ ω ∀_𝑔𝑖𝑎 = ∀_𝑔𝑗𝑢 → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
39	20, 38	jaod 855	. . . . . 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 3153	. . . . 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 479	. . . 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 855	. . 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 411	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 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 Vcvv 3472 ⊆ wss 3947 ⟨cop 4633 suc csuc 6365 ‘cfv 6542 (class class class)co 7411 ωcom 7857 2_oc2o 8462 ⊼_𝑔cgna 34623 ∀_𝑔cgol 34624 Fmlacfmla 34626

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-map 8824 df-goel 34629 df-gona 34630 df-goal 34631 df-sat 34632 df-fmla 34634

This theorem is referenced by: goalr 34686

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