Theorem gdlopcfv6

Description: The value of the combined Gödel operations function when its first argument is 6. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv6.1	$$⊢\phi \to A\in V$$
	gdlopcfv6.2	$$⊢\phi \to B\in W$$
Assertion	gdlopcfv6	Could not format assertion : No typesetting found for |- ( ph -> ( ~F ` <. 6o , A , B >. ) = ( A i^i `' B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv6.1	$$⊢\phi \to A\in V$$
2		gdlopcfv6.2	$$⊢\phi \to B\in W$$
3		6olt9o	Could not format 6o e. 9o : No typesetting found for |- 6o e. 9o with typecode |-
4		df-6o	Could not format 6o = suc 5o : No typesetting found for |- 6o = suc 5o with typecode |-
5	3 4	suceldifsn0	Could not format 6o e. ( 9o \ { (/) } ) : No typesetting found for |- 6o e. ( 9o \ { (/) } ) with typecode |-
6	5	a1i	Could not format ( ph -> 6o e. ( 9o \ { (/) } ) ) : No typesetting found for |- ( ph -> 6o 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 -> ( ~F6 ` <. A , B >. ) e. _V ) : No typesetting found for |- ( ph -> ( ~F6 ` <. A , B >. ) e. _V ) with typecode |-
10		3on	$$⊢3_{𝑜}\in \mathrm{On}$$
11		6on	Could not format 6o e. On : No typesetting found for |- 6o e. On with typecode |-
12		df-4o	$$⊢4_{𝑜}=\mathrm{suc}{3}_{𝑜}$$
13	10 12	eqsucidi	$$⊢3_{𝑜}\in 4_{𝑜}$$
14		4on	$$⊢4_{𝑜}\in \mathrm{On}$$
15		df-5o	Could not format 5o = suc 4o : No typesetting found for |- 5o = suc 4o with typecode |-
16	14 15	eqsucidi	Could not format 4o e. 5o : No typesetting found for |- 4o e. 5o with typecode |-
17		5on	Could not format 5o e. On : No typesetting found for |- 5o e. On with typecode |-
18	17 4	eqsucidi	Could not format 5o e. 6o : No typesetting found for |- 5o e. 6o with typecode |-
19	11 16 18	ontr1i	Could not format 4o e. 6o : No typesetting found for |- 4o e. 6o with typecode |-
20	11 13 19	ontr1i	Could not format 3o e. 6o : No typesetting found for |- 3o e. 6o with typecode |-
21	10 11 20	gdlopcfvlem	Could not format ( ( ph /\ 6o = 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 /\ 6o = 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 |-
22		2on	$$⊢2_{𝑜}\in \mathrm{On}$$
23		df-3o	$$⊢3_{𝑜}=\mathrm{suc}{2}_{𝑜}$$
24	22 23	eqsucidi	$$⊢2_{𝑜}\in 3_{𝑜}$$
25	11 24 20	ontr1i	Could not format 2o e. 6o : No typesetting found for |- 2o e. 6o with typecode |-
26	22 11 25	gdlopcfvlem	Could not format ( ( ph /\ 6o = 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 /\ 6o = 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 |-
27	14 11 19	gdlopcfvlem	Could not format ( ( ph /\ 6o = 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 /\ 6o = 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 |-
28	17 11 18	gdlopcfvlem	Could not format ( ( ph /\ 6o = 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 /\ 6o = 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 |-
29		eqcom	Could not format ( 6o = n <-> n = 6o ) : No typesetting found for |- ( 6o = n <-> n = 6o ) with typecode |-
30	29	bilani	Could not format ( ( ph /\ 6o = n ) -> n = 6o ) : No typesetting found for |- ( ( ph /\ 6o = n ) -> n = 6o ) with typecode |-
31	30	iftrued	Could not format ( ( ph /\ 6o = n ) -> if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) = ( ~F6 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ 6o = n ) -> if ( n = 6o , ( ~F6 ` <. x , y >. ) , if ( n = 7o , ( ~F7 ` <. x , y >. ) , ( ~F8 ` <. x , y >. ) ) ) = ( ~F6 ` <. x , y >. ) ) with typecode |-
32	27 28 31	3eqtrrd	Could not format ( ( ph /\ 6o = n ) -> ( ~F6 ` <. 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 /\ 6o = n ) -> ( ~F6 ` <. 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 |-
33	21 26 32	3eqtr4rd	Could not format ( ( ph /\ 6o = n ) -> ( ~F6 ` <. 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 /\ 6o = n ) -> ( ~F6 ` <. 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 |-
34		1on	$$⊢1_{𝑜}\in \mathrm{On}$$
35		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
36	34 35	eqsucidi	$$⊢1_{𝑜}\in 2_{𝑜}$$
37	11 36 25	ontr1i	Could not format 1o e. 6o : No typesetting found for |- 1o e. 6o with typecode |-
38	34 11 37	gdlopcfvlem	Could not format ( ( ph /\ 6o = 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 /\ 6o = 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 |-
39	33 38	eqtr4d	Could not format ( ( ph /\ 6o = n ) -> ( ~F6 ` <. 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 /\ 6o = n ) -> ( ~F6 ` <. 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 |-
40		opeq1	$$⊢A=x\to ⟨A,y⟩=⟨x,y⟩$$
41	40	fveq2d	Could not format ( A = x -> ( ~F6 ` <. A , y >. ) = ( ~F6 ` <. x , y >. ) ) : No typesetting found for |- ( A = x -> ( ~F6 ` <. A , y >. ) = ( ~F6 ` <. x , y >. ) ) with typecode |-
42	41	adantl	Could not format ( ( ph /\ A = x ) -> ( ~F6 ` <. A , y >. ) = ( ~F6 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ A = x ) -> ( ~F6 ` <. A , y >. ) = ( ~F6 ` <. x , y >. ) ) with typecode |-
43		opeq2	$$⊢B=y\to ⟨A,B⟩=⟨A,y⟩$$
44	43	fveq2d	Could not format ( B = y -> ( ~F6 ` <. A , B >. ) = ( ~F6 ` <. A , y >. ) ) : No typesetting found for |- ( B = y -> ( ~F6 ` <. A , B >. ) = ( ~F6 ` <. A , y >. ) ) with typecode |-
45	44	adantl	Could not format ( ( ph /\ B = y ) -> ( ~F6 ` <. A , B >. ) = ( ~F6 ` <. A , y >. ) ) : No typesetting found for |- ( ( ph /\ B = y ) -> ( ~F6 ` <. A , B >. ) = ( ~F6 ` <. A , y >. ) ) with typecode |-
46		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 |-
47	6 7 8 9 39 42 45 46	mpt3fvot2d	Could not format ( ph -> ( ~F ` <. 6o , A , B >. ) = ( ~F6 ` <. A , B >. ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 6o , A , B >. ) = ( ~F6 ` <. A , B >. ) ) with typecode |-
48	1 2	gdlop6fv	Could not format ( ph -> ( ~F6 ` <. A , B >. ) = ( A i^i `' B ) ) : No typesetting found for |- ( ph -> ( ~F6 ` <. A , B >. ) = ( A i^i `' B ) ) with typecode |-
49	47 48	eqtrd	Could not format ( ph -> ( ~F ` <. 6o , A , B >. ) = ( A i^i `' B ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 6o , A , B >. ) = ( A i^i `' B ) ) with typecode |-