Theorem gonarlem 36367

Description: Lemma for gonar 36368 (induction step). (Contributed by AV, 21-Oct-2023.)

Assertion

Ref	Expression
gonarlem	⊢ (𝑁 ∈ ω → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))

Distinct variable group:   𝑎,𝑏

Allowed substitution hints:   𝑁(𝑎, 𝑏)

Proof of Theorem gonarlem

Dummy variables 𝑖 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7935	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
2		ovexd 7486	. . . . 5 ⊢ (𝑁 ∈ ω → (𝑎⊼𝑔𝑏) ∈ V)
3		isfmlasuc 36361	. . . . 5 ⊢ ((suc 𝑁 ∈ ω ∧ (𝑎⊼𝑔𝑏) ∈ V) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) ↔ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢))))
4	1, 2, 3	syl2anc 596	. . . 4 ⊢ (𝑁 ∈ ω → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) ↔ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢))))
5	4	adantr 486	. . 3 ⊢ ((𝑁 ∈ ω ∧ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) ↔ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢))))
6		fmlasssuc 36362	. . . . . . . . . . 11 ⊢ (suc 𝑁 ∈ ω → (Fmla‘suc 𝑁) ⊆ (Fmla‘suc suc 𝑁))
7	1, 6	syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → (Fmla‘suc 𝑁) ⊆ (Fmla‘suc suc 𝑁))
8	7	sseld 3912	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁)))
9	7	sseld 3912	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → (𝑏 ∈ (Fmla‘suc 𝑁) → 𝑏 ∈ (Fmla‘suc suc 𝑁)))
10	8, 9	anim12d 621	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ((𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
11	10	com12 33	. . . . . . 7 ⊢ ((𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)) → (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
12	11	imim2i 17	. . . . . 6 ⊢ (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
13	12	com23 87	. . . . 5 ⊢ (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → (𝑁 ∈ ω → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
14	13	impcom 413	. . . 4 ⊢ ((𝑁 ∈ ω ∧ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
15		gonafv 36323	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ V ∧ 𝑏 ∈ V) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
16	15	el2v 3461	. . . . . . . . . . . . 13 ⊢ (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (𝑎⊼𝑔𝑏) = ⟨1o, ⟨𝑎, 𝑏⟩⟩)
18		gonafv 36323	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (𝑢⊼𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
19	17, 18	eqeq12d 2779	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ↔ ⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩))
20		1oex 8524	. . . . . . . . . . . 12 ⊢ 1o ∈ V
21		opex 5439	. . . . . . . . . . . 12 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
22	20, 21	opth 5452	. . . . . . . . . . 11 ⊢ (⟨1o, ⟨𝑎, 𝑏⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩ ↔ (1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩))
23	19, 22	bitrdi 290	. . . . . . . . . 10 ⊢ ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ↔ (1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩)))
24	23	adantll 727	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ↔ (1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩)))
25		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑎 ∈ V
26		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
27	25, 26	opth 5452	. . . . . . . . . . . . 13 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩ ↔ (𝑎 = 𝑢 ∧ 𝑏 = 𝑣))
28		eleq1w 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑎 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
29	28	equcoms 2055	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑢 → (𝑢 ∈ (Fmla‘suc 𝑁) ↔ 𝑎 ∈ (Fmla‘suc 𝑁)))
30		eleq1w 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑏 → (𝑣 ∈ (Fmla‘suc 𝑁) ↔ 𝑏 ∈ (Fmla‘suc 𝑁)))
31	30	equcoms 2055	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑣 → (𝑣 ∈ (Fmla‘suc 𝑁) ↔ 𝑏 ∈ (Fmla‘suc 𝑁)))
32	29, 31	bi2anan9 650	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) ↔ (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))))
33	32, 11	biimtrdi 256	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
34	27, 33	sylbi 220	. . . . . . . . . . . 12 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩ → ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
35	34	adantl 487	. . . . . . . . . . 11 ⊢ ((1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩) → ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → (𝑁 ∈ ω → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
36	35	com13 89	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → ((𝑢 ∈ (Fmla‘suc 𝑁) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
37	36	impl 461	. . . . . . . . 9 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((1o = 1o ∧ ⟨𝑎, 𝑏⟩ = ⟨𝑢, 𝑣⟩) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
38	24, 37	sylbid 243	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑣 ∈ (Fmla‘suc 𝑁)) → ((𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
39	38	rexlimdva 3167	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
40		gonanegoal 36325	. . . . . . . . . 10 ⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢
41		eqneqall 2970	. . . . . . . . . 10 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 → ((𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
42	40, 41	mpi 21	. . . . . . . . 9 ⊢ ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))
43	42	a1i 11	. . . . . . . 8 ⊢ (((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) ∧ 𝑖 ∈ ω) → ((𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
44	43	rexlimdva 3167	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → (∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢 → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
45	39, 44	jaod 873	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑢 ∈ (Fmla‘suc 𝑁)) → ((∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
46	45	rexlimdva 3167	. . . . 5 ⊢ (𝑁 ∈ ω → (∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
47	46	adantr 486	. . . 4 ⊢ ((𝑁 ∈ ω ∧ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)))) → (∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
48	14, 47	jaod 873	. . 3 ⊢ ((𝑁 ∈ ω ∧ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)))) → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) ∨ ∃𝑢 ∈ (Fmla‘suc 𝑁)(∃𝑣 ∈ (Fmla‘suc 𝑁)(𝑎⊼𝑔𝑏) = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω (𝑎⊼𝑔𝑏) = ∀𝑔𝑖𝑢)) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
49	5, 48	sylbid 243	. 2 ⊢ ((𝑁 ∈ ω ∧ ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁)))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁))))
50	49	ex 418	1 ⊢ (𝑁 ∈ ω → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ⟨cop 4840  suc csuc 6390  ‘cfv 6566  (class class class)co 7451  ωcom 7911  1_oc1o 8507  ⊼_𝑔cgna 36307  ∀_𝑔cgol 36308  Fmlacfmla 36310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-map 8887  df-goel 36313  df-gona 36314  df-goal 36315  df-sat 36316  df-fmla 36318

This theorem is used by:  gonar  36368