Theorem gdlopcfv7

Description: The value of the combined Gödel operations function when its first argument is 7. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv7.1	$$⊢\phi \to A\in V$$
	gdlopcfv7.2	$$⊢\phi \to B\in W$$
Assertion	gdlopcfv7	Could not format assertion : No typesetting found for |- ( ph -> ( ~F ` <. 7o , A , B >. ) = ( A i^i ( Cnv2 ` B ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv7.1	$$⊢\phi \to A\in V$$
2		gdlopcfv7.2	$$⊢\phi \to B\in W$$
3		7olt9o	Could not format 7o e. 9o : No typesetting found for |- 7o e. 9o with typecode |-
4		df-7o	Could not format 7o = suc 6o : No typesetting found for |- 7o = suc 6o with typecode |-
5	3 4	suceldifsn0	Could not format 7o e. ( 9o \ { (/) } ) : No typesetting found for |- 7o e. ( 9o \ { (/) } ) with typecode |-
6	5	a1i	Could not format ( ph -> 7o e. ( 9o \ { (/) } ) ) : No typesetting found for |- ( ph -> 7o e. ( 9o \ { (/) } ) ) with typecode |-
7	1	elexd	$$⊢\phi \to A\in \mathrm{V}$$
8	2	elexd	$$⊢\phi \to B\in \mathrm{V}$$
9		fvexd	Could not format ( ph -> ( ~F7 ` <. A , B >. ) e. _V ) : No typesetting found for |- ( ph -> ( ~F7 ` <. A , B >. ) e. _V ) with typecode |-
10		2on	$$⊢2_{𝑜}\in \mathrm{On}$$
11		7on	Could not format 7o e. On : No typesetting found for |- 7o e. On with typecode |-
12		df-3o	$$⊢3_{𝑜}=\mathrm{suc}{2}_{𝑜}$$
13	10 12	eqsucidi	$$⊢2_{𝑜}\in 3_{𝑜}$$
14		3on	$$⊢3_{𝑜}\in \mathrm{On}$$
15		df-4o	$$⊢4_{𝑜}=\mathrm{suc}{3}_{𝑜}$$
16	14 15	eqsucidi	$$⊢3_{𝑜}\in 4_{𝑜}$$
17		4on	$$⊢4_{𝑜}\in \mathrm{On}$$
18		df-5o	Could not format 5o = suc 4o : No typesetting found for |- 5o = suc 4o with typecode |-
19	17 18	eqsucidi	Could not format 4o e. 5o : No typesetting found for |- 4o e. 5o with typecode |-
20		5on	Could not format 5o e. On : No typesetting found for |- 5o e. On with typecode |-
21		df-6o	Could not format 6o = suc 5o : No typesetting found for |- 6o = suc 5o with typecode |-
22	20 21	eqsucidi	Could not format 5o e. 6o : No typesetting found for |- 5o e. 6o with typecode |-
23		6on	Could not format 6o e. On : No typesetting found for |- 6o e. On with typecode |-
24	23 4	eqsucidi	Could not format 6o e. 7o : No typesetting found for |- 6o e. 7o with typecode |-
25	11 22 24	ontr1i	Could not format 5o e. 7o : No typesetting found for |- 5o e. 7o with typecode |-
26	11 19 25	ontr1i	Could not format 4o e. 7o : No typesetting found for |- 4o e. 7o with typecode |-
27	11 16 26	ontr1i	Could not format 3o e. 7o : No typesetting found for |- 3o e. 7o with typecode |-
28	11 13 27	ontr1i	Could not format 2o e. 7o : No typesetting found for |- 2o e. 7o with typecode |-
29	10 11 28	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) = if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) = if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) with typecode |-
30		1on	$$⊢1_{𝑜}\in \mathrm{On}$$
31		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
32	30 31	eqsucidi	$$⊢1_{𝑜}\in 2_{𝑜}$$
33	11 32 28	ontr1i	Could not format 1o e. 7o : No typesetting found for |- 1o e. 7o with typecode |-
34	30 11 33	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) = if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) = if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) with typecode |-
35	17 11 26	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) = if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) = if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) with typecode |-
36	14 11 27	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) = if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) = if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) with typecode |-
37	20 11 25	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) = if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) = if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) with typecode |-
38	23 11 24	gdlopcfvlem	Could not format ( ( ph /\ 7o = n ) -> if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) = if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) = if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) with typecode |-
39		eqcom	Could not format ( 7o = n <-> n = 7o ) : No typesetting found for |- ( 7o = n <-> n = 7o ) with typecode |-
40	39	bilani	Could not format ( ( ph /\ 7o = n ) -> n = 7o ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> n = 7o ) with typecode |-
41	40	iftrued	Could not format ( ( ph /\ 7o = n ) -> if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) = ( ~F7 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) = ( ~F7 ` <. x , y >. ) ) with typecode |-
42	37 38 41	3eqtrrd	Could not format ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) with typecode |-
43	35 36 42	3eqtr4rd	Could not format ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) with typecode |-
44	29 34 43	3eqtr4rd	Could not format ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) ) : No typesetting found for |- ( ( ph /\ 7o = n ) -> ( ~F7 ` <. x , y >. ) = if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) ) with typecode |-
45		opeq1	$$⊢A=x\to ⟨A,y⟩=⟨x,y⟩$$
46	45	fveq2d	Could not format ( A = x -> ( ~F7 ` <. A , y >. ) = ( ~F7 ` <. x , y >. ) ) : No typesetting found for |- ( A = x -> ( ~F7 ` <. A , y >. ) = ( ~F7 ` <. x , y >. ) ) with typecode |-
47	46	adantl	Could not format ( ( ph /\ A = x ) -> ( ~F7 ` <. A , y >. ) = ( ~F7 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ A = x ) -> ( ~F7 ` <. A , y >. ) = ( ~F7 ` <. x , y >. ) ) with typecode |-
48		opeq2	$$⊢B=y\to ⟨A,B⟩=⟨A,y⟩$$
49	48	fveq2d	Could not format ( B = y -> ( ~F7 ` <. A , B >. ) = ( ~F7 ` <. A , y >. ) ) : No typesetting found for |- ( B = y -> ( ~F7 ` <. A , B >. ) = ( ~F7 ` <. A , y >. ) ) with typecode |-
50	49	adantl	Could not format ( ( ph /\ B = y ) -> ( ~F7 ` <. A , B >. ) = ( ~F7 ` <. A , y >. ) ) : No typesetting found for |- ( ( ph /\ B = y ) -> ( ~F7 ` <. A , B >. ) = ( ~F7 ` <. A , y >. ) ) with typecode |-
51		df-gdlopc	Could not format ~F = ( n e. ( 9o \ { (/) } ) , x e. _V , y e. _V |-> if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) ) : No typesetting found for |- ~F = ( n e. ( 9o \ { (/) } ) , x e. _V , y e. _V |-> if ( n = 1o , ( ~F1 ` <. x , y >. ) , if ( n = 2o , ( ~F2 ` <. x , y >. ) , if ( n = 3o , ( ~F3 ` <. x , y >. ) , if ( n = 4o , ( ~F4 ` <. x , y >. ) , if ( n = 5o , ( ~F5 ` <. x , y >. ) , if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) ) ) ) ) ) ) with typecode |-
52	6 7 8 9 44 47 50 51	mpt3fvot2d	Could not format ( ph -> ( ~F ` <. 7o , A , B >. ) = ( ~F7 ` <. A , B >. ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 7o , A , B >. ) = ( ~F7 ` <. A , B >. ) ) with typecode |-
53	1 2	gdlop7fv	Could not format ( ph -> ( ~F7 ` <. A , B >. ) = ( A i^i ( Cnv2 ` B ) ) ) : No typesetting found for |- ( ph -> ( ~F7 ` <. A , B >. ) = ( A i^i ( Cnv2 ` B ) ) ) with typecode |-
54	52 53	eqtrd	Could not format ( ph -> ( ~F ` <. 7o , A , B >. ) = ( A i^i ( Cnv2 ` B ) ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 7o , A , B >. ) = ( A i^i ( Cnv2 ` B ) ) ) with typecode |-