Theorem 2lgslem1b 15811

Description: Lemma 2 for 2lgslem1 15813. (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 2236	. . . . . 6 |- ( x = ( j x. 2 ) -> ( x = ( i x. 2 ) <-> ( j x. 2 ) = ( i x. 2 ) ) )
3	2	rexbidv 2531	. . . . 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 10253	. . . . . . 7 |- ( j e. ( A ... B ) -> j e. ZZ )
5		2lgslem1b.i	. . . . . . 7 |- I = ( A ... B )
6	4, 5	eleq2s 2324	. . . . . 6 |- ( j e. I -> j e. ZZ )
7		2z 9500	. . . . . . 7 |- 2 e. ZZ
8	7	a1i 9	. . . . . 6 |- ( j e. I -> 2 e. ZZ )
9	6, 8	zmulcld 9601	. . . . 5 |- ( j e. I -> ( j x. 2 ) e. ZZ )
10		id 19	. . . . . 6 |- ( j e. I -> j e. I )
11		oveq1 6020	. . . . . . . 8 |- ( i = j -> ( i x. 2 ) = ( j x. 2 ) )
12	11	eqeq2d 2241	. . . . . . 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 2230	. . . . . 6 |- ( j e. I -> ( j x. 2 ) = ( j x. 2 ) )
15	10, 13, 14	rspcedvd 2914	. . . . 5 |- ( j e. I -> E. i e. I ( j x. 2 ) = ( i x. 2 ) )
16	3, 9, 15	elrabd 2962	. . . 4 |- ( j e. I -> ( j x. 2 ) e. { x e. ZZ | E. i e. I x = ( i x. 2 ) } )
17	1, 16	fmpti 5795	. . 3 |- F : I --> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
18		oveq1 6020	. . . . . . 7 |- ( j = y -> ( j x. 2 ) = ( y x. 2 ) )
19		simpl 109	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. I )
20		elfzelz 10253	. . . . . . . . . 10 |- ( y e. ( A ... B ) -> y e. ZZ )
21	20, 5	eleq2s 2324	. . . . . . . . 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 9601	. . . . . . . . 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 5736	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` y ) = ( y x. 2 ) )
28		oveq1 6020	. . . . . . 7 |- ( j = z -> ( j x. 2 ) = ( z x. 2 ) )
29		simpr 110	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. I )
30		elfzelz 10253	. . . . . . . . . 10 |- ( z e. ( A ... B ) -> z e. ZZ )
31	30, 5	eleq2s 2324	. . . . . . . . 9 |- ( z e. I -> z e. ZZ )
32	7	a1i 9	. . . . . . . . 9 |- ( z e. I -> 2 e. ZZ )
33	31, 32	zmulcld 9601	. . . . . . . 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 5736	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` z ) = ( z x. 2 ) )
36	27, 35	eqeq12d 2244	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) <-> ( y x. 2 ) = ( z x. 2 ) ) )
37	21	zcnd 9596	. . . . . . . 8 |- ( y e. I -> y e. CC )
38	37	adantr 276	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. CC )
39	31	zcnd 9596	. . . . . . . 8 |- ( z e. I -> z e. CC )
40	39	adantl 277	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. CC )
41		2cnd 9209	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 e. CC )
42		2ap0 9229	. . . . . . . 8 |- 2 # 0
43	42	a1i 9	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 # 0 )
44	38, 40, 41, 43	mulcanap2d 8835	. . . . . 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 2616	. . 3 |- A. y e. I A. z e. I ( ( F ` y ) = ( F ` z ) -> y = z )
48		dff13 5904	. . 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 948	. 2 |- F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
50		oveq1 6020	. . . . . . 7 |- ( j = i -> ( j x. 2 ) = ( i x. 2 ) )
51	50	eqeq2d 2241	. . . . . 6 |- ( j = i -> ( x = ( j x. 2 ) <-> x = ( i x. 2 ) ) )
52	51	cbvrexvw 2770	. . . . 5 |- ( E. j e. I x = ( j x. 2 ) <-> E. i e. I x = ( i x. 2 ) )
53		elfzelz 10253	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> i e. ZZ )
54	7	a1i 9	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> 2 e. ZZ )
55	53, 54	zmulcld 9601	. . . . . . . . 9 |- ( i e. ( A ... B ) -> ( i x. 2 ) e. ZZ )
56	55, 5	eleq2s 2324	. . . . . . . 8 |- ( i e. I -> ( i x. 2 ) e. ZZ )
57		eleq1 2292	. . . . . . . 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 2642	. . . . . 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 2345	. . 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 4978	. . 3 |- ran F = { x | E. j e. I x = ( j x. 2 ) }
64		df-rab 2517	. . 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 2260	. 2 |- ran F = { x e. ZZ | E. i e. I x = ( i x. 2 ) }
66		dff1o5 5589	. 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 948	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   {crab 2512   class class class wbr 4086   |-> cmpt 4148   ran crn 4724   -->wf 5320   -1-1->wf1 5321   -1-1-onto->wf1o 5323   ` cfv 5324  (class class class)co 6013   CCcc 8023   0cc0 8025   x. cmul 8030   # cap 8754   2c2 9187   ZZcz 9472   ...cfz 10236

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-inn 9137  df-2 9195  df-n0 9396  df-z 9473  df-uz 9749  df-fz 10237

This theorem is referenced by:  2lgslem1  15813