Theorem gdlopcfv4

Description: The value of the combined Gödel operations function when its first argument is 4. (Contributed by BTernaryTau, 19-Oct-2026)

	Ref	Expression
Hypotheses	gdlopcfv4.1	$$⊢\phi \to A\in V$$
	gdlopcfv4.2	$$⊢\phi \to B\in W$$
Assertion	gdlopcfv4	Could not format assertion : No typesetting found for |- ( ph -> ( ~F ` <. 4o , A , B >. ) = ( A |` B ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		gdlopcfv4.1	$$⊢\phi \to A\in V$$
2		gdlopcfv4.2	$$⊢\phi \to B\in W$$
3		4olt9o	Could not format 4o e. 9o : No typesetting found for |- 4o e. 9o with typecode |-
4		df-4o	$$⊢4_{𝑜}=\mathrm{suc}{3}_{𝑜}$$
5	3 4	suceldifsn0	Could not format 4o e. ( 9o \ { (/) } ) : No typesetting found for |- 4o e. ( 9o \ { (/) } ) with typecode |-
6	5	a1i	Could not format ( ph -> 4o e. ( 9o \ { (/) } ) ) : No typesetting found for |- ( ph -> 4o 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 -> ( ~F4 ` <. A , B >. ) e. _V ) : No typesetting found for |- ( ph -> ( ~F4 ` <. A , B >. ) e. _V ) with typecode |-
10		2on	$$⊢2_{𝑜}\in \mathrm{On}$$
11		4on	$$⊢4_{𝑜}\in \mathrm{On}$$
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	14 4	eqsucidi	$$⊢3_{𝑜}\in 4_{𝑜}$$
16	11 13 15	ontr1i	$$⊢2_{𝑜}\in 4_{𝑜}$$
17	10 11 16	gdlopcfvlem	Could not format ( ( ph /\ 4o = 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 /\ 4o = 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 |-
18	14 11 15	gdlopcfvlem	Could not format ( ( ph /\ 4o = 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 /\ 4o = 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 |-
19		eqcom	$$⊢4_{𝑜}=n\leftrightarrow n=4_{𝑜}$$
20	19	bilani	$$⊢\left(\phi \wedge 4_{𝑜}=n\right)\to n=4_{𝑜}$$
21	20	iftrued	Could not format ( ( ph /\ 4o = 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 >. ) ) ) ) ) = ( ~F4 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ 4o = 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 >. ) ) ) ) ) = ( ~F4 ` <. x , y >. ) ) with typecode |-
22	17 18 21	3eqtrrd	Could not format ( ( ph /\ 4o = n ) -> ( ~F4 ` <. 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 /\ 4o = n ) -> ( ~F4 ` <. 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 |-
23		1on	$$⊢1_{𝑜}\in \mathrm{On}$$
24		df-2o	$$⊢2_{𝑜}=\mathrm{suc}{1}_{𝑜}$$
25	23 24	eqsucidi	$$⊢1_{𝑜}\in 2_{𝑜}$$
26	14 25 13	ontr1i	$$⊢1_{𝑜}\in 3_{𝑜}$$
27	11 26 15	ontr1i	$$⊢1_{𝑜}\in 4_{𝑜}$$
28	23 11 27	gdlopcfvlem	Could not format ( ( ph /\ 4o = 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 /\ 4o = 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 |-
29	22 28	eqtr4d	Could not format ( ( ph /\ 4o = n ) -> ( ~F4 ` <. 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 /\ 4o = n ) -> ( ~F4 ` <. 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 |-
30		opeq1	$$⊢A=x\to ⟨A,y⟩=⟨x,y⟩$$
31	30	fveq2d	Could not format ( A = x -> ( ~F4 ` <. A , y >. ) = ( ~F4 ` <. x , y >. ) ) : No typesetting found for |- ( A = x -> ( ~F4 ` <. A , y >. ) = ( ~F4 ` <. x , y >. ) ) with typecode |-
32	31	adantl	Could not format ( ( ph /\ A = x ) -> ( ~F4 ` <. A , y >. ) = ( ~F4 ` <. x , y >. ) ) : No typesetting found for |- ( ( ph /\ A = x ) -> ( ~F4 ` <. A , y >. ) = ( ~F4 ` <. x , y >. ) ) with typecode |-
33		opeq2	$$⊢B=y\to ⟨A,B⟩=⟨A,y⟩$$
34	33	fveq2d	Could not format ( B = y -> ( ~F4 ` <. A , B >. ) = ( ~F4 ` <. A , y >. ) ) : No typesetting found for |- ( B = y -> ( ~F4 ` <. A , B >. ) = ( ~F4 ` <. A , y >. ) ) with typecode |-
35	34	adantl	Could not format ( ( ph /\ B = y ) -> ( ~F4 ` <. A , B >. ) = ( ~F4 ` <. A , y >. ) ) : No typesetting found for |- ( ( ph /\ B = y ) -> ( ~F4 ` <. A , B >. ) = ( ~F4 ` <. A , y >. ) ) with typecode |-
36		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 |-
37	6 7 8 9 29 32 35 36	mpt3fvot2d	Could not format ( ph -> ( ~F ` <. 4o , A , B >. ) = ( ~F4 ` <. A , B >. ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 4o , A , B >. ) = ( ~F4 ` <. A , B >. ) ) with typecode |-
38	1 2	gdlop4fv	Could not format ( ph -> ( ~F4 ` <. A , B >. ) = ( A |` B ) ) : No typesetting found for |- ( ph -> ( ~F4 ` <. A , B >. ) = ( A |` B ) ) with typecode |-
39	37 38	eqtrd	Could not format ( ph -> ( ~F ` <. 4o , A , B >. ) = ( A |` B ) ) : No typesetting found for |- ( ph -> ( ~F ` <. 4o , A , B >. ) = ( A |` B ) ) with typecode |-