Theorem 2lgslem1b 15949

Description: Lemma 2 for 2lgslem1 15951. (Contributed by AV, 18-Jun-2021.)

Hypotheses

Ref	Expression
2lgslem1b.i	|- I = ( A ... B )
2lgslem1b.f	|- F = ( j e. I |-> ( j x. 2 ) )

Assertion

Ref	Expression
2lgslem1b	|- F : I -1-1-onto-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }

Distinct variable group:   i, I, j, x

Allowed substitution hints:   A( x, i, j)   B( x, i, j)   F( x, i, j)

Proof of Theorem 2lgslem1b

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2lgslem1b.f	. . . 4 |- F = ( j e. I |-> ( j x. 2 ) )
2		eqeq1 2239	. . . . . 6 |- ( x = ( j x. 2 ) -> ( x = ( i x. 2 ) <-> ( j x. 2 ) = ( i x. 2 ) ) )
3	2	rexbidv 2543	. . . . 5 |- ( x = ( j x. 2 ) -> ( E. i e. I x = ( i x. 2 ) <-> E. i e. I ( j x. 2 ) = ( i x. 2 ) ) )
4		elfzelz 10355	. . . . . . 7 |- ( j e. ( A ... B ) -> j e. ZZ )
5		2lgslem1b.i	. . . . . . 7 |- I = ( A ... B )
6	4, 5	eleq2s 2327	. . . . . 6 |- ( j e. I -> j e. ZZ )
7		2z 9601	. . . . . . 7 |- 2 e. ZZ
8	7	a1i 9	. . . . . 6 |- ( j e. I -> 2 e. ZZ )
9	6, 8	zmulcld 9702	. . . . 5 |- ( j e. I -> ( j x. 2 ) e. ZZ )
10		id 19	. . . . . 6 |- ( j e. I -> j e. I )
11		oveq1 6056	. . . . . . . 8 |- ( i = j -> ( i x. 2 ) = ( j x. 2 ) )
12	11	eqeq2d 2244	. . . . . . 7 |- ( i = j -> ( ( j x. 2 ) = ( i x. 2 ) <-> ( j x. 2 ) = ( j x. 2 ) ) )
13	12	adantl 277	. . . . . 6 |- ( ( j e. I /\ i = j ) -> ( ( j x. 2 ) = ( i x. 2 ) <-> ( j x. 2 ) = ( j x. 2 ) ) )
14		eqidd 2233	. . . . . 6 |- ( j e. I -> ( j x. 2 ) = ( j x. 2 ) )
15	10, 13, 14	rspcedvd 2926	. . . . 5 |- ( j e. I -> E. i e. I ( j x. 2 ) = ( i x. 2 ) )
16	3, 9, 15	elrabd 2974	. . . 4 |- ( j e. I -> ( j x. 2 ) e. { x e. ZZ | E. i e. I x = ( i x. 2 ) } )
17	1, 16	fmpti 5828	. . 3 |- F : I --> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
18		oveq1 6056	. . . . . . 7 |- ( j = y -> ( j x. 2 ) = ( y x. 2 ) )
19		simpl 109	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. I )
20		elfzelz 10355	. . . . . . . . . 10 |- ( y e. ( A ... B ) -> y e. ZZ )
21	20, 5	eleq2s 2327	. . . . . . . . 9 |- ( y e. I -> y e. ZZ )
22		id 19	. . . . . . . . . 10 |- ( y e. ZZ -> y e. ZZ )
23	7	a1i 9	. . . . . . . . . 10 |- ( y e. ZZ -> 2 e. ZZ )
24	22, 23	zmulcld 9702	. . . . . . . . 9 |- ( y e. ZZ -> ( y x. 2 ) e. ZZ )
25	21, 24	syl 14	. . . . . . . 8 |- ( y e. I -> ( y x. 2 ) e. ZZ )
26	25	adantr 276	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> ( y x. 2 ) e. ZZ )
27	1, 18, 19, 26	fvmptd3 5770	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` y ) = ( y x. 2 ) )
28		oveq1 6056	. . . . . . 7 |- ( j = z -> ( j x. 2 ) = ( z x. 2 ) )
29		simpr 110	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. I )
30		elfzelz 10355	. . . . . . . . . 10 |- ( z e. ( A ... B ) -> z e. ZZ )
31	30, 5	eleq2s 2327	. . . . . . . . 9 |- ( z e. I -> z e. ZZ )
32	7	a1i 9	. . . . . . . . 9 |- ( z e. I -> 2 e. ZZ )
33	31, 32	zmulcld 9702	. . . . . . . 8 |- ( z e. I -> ( z x. 2 ) e. ZZ )
34	33	adantl 277	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> ( z x. 2 ) e. ZZ )
35	1, 28, 29, 34	fvmptd3 5770	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` z ) = ( z x. 2 ) )
36	27, 35	eqeq12d 2247	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) <-> ( y x. 2 ) = ( z x. 2 ) ) )
37	21	zcnd 9697	. . . . . . . 8 |- ( y e. I -> y e. CC )
38	37	adantr 276	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. CC )
39	31	zcnd 9697	. . . . . . . 8 |- ( z e. I -> z e. CC )
40	39	adantl 277	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. CC )
41		2cnd 9306	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 e. CC )
42		2ap0 9326	. . . . . . . 8 |- 2 # 0
43	42	a1i 9	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 # 0 )
44	38, 40, 41, 43	mulcanap2d 8932	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( ( y x. 2 ) = ( z x. 2 ) <-> y = z ) )
45	44	biimpd 144	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( y x. 2 ) = ( z x. 2 ) -> y = z ) )
46	36, 45	sylbid 150	. . . 4 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) -> y = z ) )
47	46	rgen2 2628	. . 3 |- A. y e. I A. z e. I ( ( F ` y ) = ( F ` z ) -> y = z )
48		dff13 5940	. . 3 |- ( F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) } <-> ( F : I --> { x e. ZZ | E. i e. I x = ( i x. 2 ) } /\ A. y e. I A. z e. I ( ( F ` y ) = ( F ` z ) -> y = z ) ) )
49	17, 47, 48	mpbir2an 951	. 2 |- F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
50		oveq1 6056	. . . . . . 7 |- ( j = i -> ( j x. 2 ) = ( i x. 2 ) )
51	50	eqeq2d 2244	. . . . . 6 |- ( j = i -> ( x = ( j x. 2 ) <-> x = ( i x. 2 ) ) )
52	51	cbvrexvw 2782	. . . . 5 |- ( E. j e. I x = ( j x. 2 ) <-> E. i e. I x = ( i x. 2 ) )
53		elfzelz 10355	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> i e. ZZ )
54	7	a1i 9	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> 2 e. ZZ )
55	53, 54	zmulcld 9702	. . . . . . . . 9 |- ( i e. ( A ... B ) -> ( i x. 2 ) e. ZZ )
56	55, 5	eleq2s 2327	. . . . . . . 8 |- ( i e. I -> ( i x. 2 ) e. ZZ )
57		eleq1 2295	. . . . . . . 8 |- ( x = ( i x. 2 ) -> ( x e. ZZ <-> ( i x. 2 ) e. ZZ ) )
58	56, 57	syl5ibrcom 157	. . . . . . 7 |- ( i e. I -> ( x = ( i x. 2 ) -> x e. ZZ ) )
59	58	rexlimiv 2654	. . . . . 6 |- ( E. i e. I x = ( i x. 2 ) -> x e. ZZ )
60	59	pm4.71ri 392	. . . . 5 |- ( E. i e. I x = ( i x. 2 ) <-> ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) )
61	52, 60	bitri 184	. . . 4 |- ( E. j e. I x = ( j x. 2 ) <-> ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) )
62	61	abbii 2348	. . 3 |- { x | E. j e. I x = ( j x. 2 ) } = { x | ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) }
63	1	rnmpt 5004	. . 3 |- ran F = { x | E. j e. I x = ( j x. 2 ) }
64		df-rab 2529	. . 3 |- { x e. ZZ | E. i e. I x = ( i x. 2 ) } = { x | ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) }
65	62, 63, 64	3eqtr4i 2263	. 2 |- ran F = { x e. ZZ | E. i e. I x = ( i x. 2 ) }
66		dff1o5 5622	. 2 |- ( F : I -1-1-onto-> { x e. ZZ | E. i e. I x = ( i x. 2 ) } <-> ( F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) } /\ ran F = { x e. ZZ | E. i e. I x = ( i x. 2 ) } ) )
67	49, 65, 66	mpbir2an 951	1 |- F : I -1-1-onto-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   {crab 2524   class class class wbr 4108   |-> cmpt 4170   ran crn 4749   -->wf 5347   -1-1->wf1 5348   -1-1-onto->wf1o 5350   ` cfv 5351  (class class class)co 6049   CCcc 8121   0cc0 8123   x. cmul 8128   # cap 8851   2c2 9284   ZZcz 9573   ...cfz 10338

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-po 4416  df-iso 4417  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339

This theorem is referenced by:  2lgslem1  15951