Theorem nninfwlporlemd 7238

Description: Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)

Hypotheses

Ref	Expression
nninfwlporlem.x	|- ( ph -> X : om --> 2o )
nninfwlporlem.y	|- ( ph -> Y : om --> 2o )
nninfwlporlem.d	|- D = ( i e. om |-> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )

Assertion

Ref	Expression
nninfwlporlemd	|- ( ph -> ( X = Y <-> D = ( i e. om |-> 1o ) ) )

Distinct variable groups:   D, i   i, X   i, Y   ph, i

Proof of Theorem nninfwlporlemd

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 6490	. . . . . . . . 9 |- 1o =/= (/)
2	1	neii 2369	. . . . . . . 8 |- -. 1o = (/)
3	2	intnan 930	. . . . . . 7 |- -. ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) )
4	3	biorfi 747	. . . . . 6 |- ( ( X ` i ) = ( Y ` i ) <-> ( ( X ` i ) = ( Y ` i ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) )
5		eqid 2196	. . . . . . . 8 |- 1o = 1o
6	5	biantru 302	. . . . . . 7 |- ( ( X ` i ) = ( Y ` i ) <-> ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) )
7	6	orbi1i 764	. . . . . 6 |- ( ( ( X ` i ) = ( Y ` i ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) )
8	4, 7	bitri 184	. . . . 5 |- ( ( X ` i ) = ( Y ` i ) <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) )
9		eqcom 2198	. . . . . 6 |- ( 1o = ( D ` i ) <-> ( D ` i ) = 1o )
10		nninfwlporlem.d	. . . . . . . . . 10 |- D = ( i e. om |-> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )
11		fveq2 5558	. . . . . . . . . . . . 13 |- ( i = j -> ( X ` i ) = ( X ` j ) )
12		fveq2 5558	. . . . . . . . . . . . 13 |- ( i = j -> ( Y ` i ) = ( Y ` j ) )
13	11, 12	eqeq12d 2211	. . . . . . . . . . . 12 |- ( i = j -> ( ( X ` i ) = ( Y ` i ) <-> ( X ` j ) = ( Y ` j ) ) )
14	13	ifbid 3582	. . . . . . . . . . 11 |- ( i = j -> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) = if ( ( X ` j ) = ( Y ` j ) , 1o , (/) ) )
15	14	cbvmptv 4129	. . . . . . . . . 10 |- ( i e. om |-> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) ) = ( j e. om |-> if ( ( X ` j ) = ( Y ` j ) , 1o , (/) ) )
16	10, 15	eqtri 2217	. . . . . . . . 9 |- D = ( j e. om |-> if ( ( X ` j ) = ( Y ` j ) , 1o , (/) ) )
17		fveq2 5558	. . . . . . . . . . 11 |- ( j = i -> ( X ` j ) = ( X ` i ) )
18		fveq2 5558	. . . . . . . . . . 11 |- ( j = i -> ( Y ` j ) = ( Y ` i ) )
19	17, 18	eqeq12d 2211	. . . . . . . . . 10 |- ( j = i -> ( ( X ` j ) = ( Y ` j ) <-> ( X ` i ) = ( Y ` i ) ) )
20	19	ifbid 3582	. . . . . . . . 9 |- ( j = i -> if ( ( X ` j ) = ( Y ` j ) , 1o , (/) ) = if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )
21		simpr 110	. . . . . . . . 9 |- ( ( ph /\ i e. om ) -> i e. om )
22		1lt2o 6500	. . . . . . . . . . 11 |- 1o e. 2o
23	22	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ i e. om ) -> 1o e. 2o )
24		0lt2o 6499	. . . . . . . . . . 11 |- (/) e. 2o
25	24	a1i 9	. . . . . . . . . 10 |- ( ( ph /\ i e. om ) -> (/) e. 2o )
26		2ssom 6582	. . . . . . . . . . . 12 |- 2o C_ om
27		nninfwlporlem.x	. . . . . . . . . . . . 13 |- ( ph -> X : om --> 2o )
28	27	ffvelcdmda 5697	. . . . . . . . . . . 12 |- ( ( ph /\ i e. om ) -> ( X ` i ) e. 2o )
29	26, 28	sselid 3181	. . . . . . . . . . 11 |- ( ( ph /\ i e. om ) -> ( X ` i ) e. om )
30		nninfwlporlem.y	. . . . . . . . . . . . 13 |- ( ph -> Y : om --> 2o )
31	30	ffvelcdmda 5697	. . . . . . . . . . . 12 |- ( ( ph /\ i e. om ) -> ( Y ` i ) e. 2o )
32	26, 31	sselid 3181	. . . . . . . . . . 11 |- ( ( ph /\ i e. om ) -> ( Y ` i ) e. om )
33		nndceq 6557	. . . . . . . . . . 11 |- ( ( ( X ` i ) e. om /\ ( Y ` i ) e. om ) -> DECID ( X ` i ) = ( Y ` i ) )
34	29, 32, 33	syl2anc 411	. . . . . . . . . 10 |- ( ( ph /\ i e. om ) -> DECID ( X ` i ) = ( Y ` i ) )
35	23, 25, 34	ifcldcd 3597	. . . . . . . . 9 |- ( ( ph /\ i e. om ) -> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) e. 2o )
36	16, 20, 21, 35	fvmptd3 5655	. . . . . . . 8 |- ( ( ph /\ i e. om ) -> ( D ` i ) = if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )
37	36	eqeq2d 2208	. . . . . . 7 |- ( ( ph /\ i e. om ) -> ( 1o = ( D ` i ) <-> 1o = if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) ) )
38		eqifdc 3596	. . . . . . . 8 |- (DECID ( X ` i ) = ( Y ` i ) -> ( 1o = if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) ) )
39	34, 38	syl 14	. . . . . . 7 |- ( ( ph /\ i e. om ) -> ( 1o = if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) ) )
40	37, 39	bitrd 188	. . . . . 6 |- ( ( ph /\ i e. om ) -> ( 1o = ( D ` i ) <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) ) )
41	9, 40	bitr3id 194	. . . . 5 |- ( ( ph /\ i e. om ) -> ( ( D ` i ) = 1o <-> ( ( ( X ` i ) = ( Y ` i ) /\ 1o = 1o ) \/ ( -. ( X ` i ) = ( Y ` i ) /\ 1o = (/) ) ) ) )
42	8, 41	bitr4id 199	. . . 4 |- ( ( ph /\ i e. om ) -> ( ( X ` i ) = ( Y ` i ) <-> ( D ` i ) = 1o ) )
43	42	ralbidva 2493	. . 3 |- ( ph -> ( A. i e. om ( X ` i ) = ( Y ` i ) <-> A. i e. om ( D ` i ) = 1o ) )
44		fveqeq2 5567	. . . 4 |- ( i = j -> ( ( D ` i ) = 1o <-> ( D ` j ) = 1o ) )
45	44	cbvralv 2729	. . 3 |- ( A. i e. om ( D ` i ) = 1o <-> A. j e. om ( D ` j ) = 1o )
46	43, 45	bitrdi 196	. 2 |- ( ph -> ( A. i e. om ( X ` i ) = ( Y ` i ) <-> A. j e. om ( D ` j ) = 1o ) )
47	27	ffnd 5408	. . 3 |- ( ph -> X Fn om )
48	30	ffnd 5408	. . 3 |- ( ph -> Y Fn om )
49		eqfnfv 5659	. . 3 |- ( ( X Fn om /\ Y Fn om ) -> ( X = Y <-> A. i e. om ( X ` i ) = ( Y ` i ) ) )
50	47, 48, 49	syl2anc 411	. 2 |- ( ph -> ( X = Y <-> A. i e. om ( X ` i ) = ( Y ` i ) ) )
51	35	ralrimiva 2570	. . . 4 |- ( ph -> A. i e. om if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) e. 2o )
52	10	fnmpt 5384	. . . 4 |- ( A. i e. om if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) e. 2o -> D Fn om )
53	51, 52	syl 14	. . 3 |- ( ph -> D Fn om )
54		eqidd 2197	. . 3 |- ( j = i -> 1o = 1o )
55		1onn 6578	. . . 4 |- 1o e. om
56	55	a1i 9	. . 3 |- ( ( ph /\ j e. om ) -> 1o e. om )
57	55	a1i 9	. . 3 |- ( ( ph /\ i e. om ) -> 1o e. om )
58	53, 54, 56, 57	fnmptfvd 5666	. 2 |- ( ph -> ( D = ( i e. om |-> 1o ) <-> A. j e. om ( D ` j ) = 1o ) )
59	46, 50, 58	3bitr4d 220	1 |- ( ph -> ( X = Y <-> D = ( i e. om |-> 1o ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   A.wral 2475   (/)c0 3450   ifcif 3561   |-> cmpt 4094   omcom 4626   Fn wfn 5253   -->wf 5254   ` cfv 5258   1oc1o 6467   2oc2o 6468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-1o 6474  df-2o 6475

This theorem is referenced by:  nninfwlporlem  7239