Theorem 2lgslem1b 15821

Description: Lemma 2 for 2lgslem1 15823. (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 2238	. . . . . 6 |- ( x = ( j x. 2 ) -> ( x = ( i x. 2 ) <-> ( j x. 2 ) = ( i x. 2 ) ) )
3	2	rexbidv 2533	. . . . 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 10260	. . . . . . 7 |- ( j e. ( A ... B ) -> j e. ZZ )
5		2lgslem1b.i	. . . . . . 7 |- I = ( A ... B )
6	4, 5	eleq2s 2326	. . . . . 6 |- ( j e. I -> j e. ZZ )
7		2z 9507	. . . . . . 7 |- 2 e. ZZ
8	7	a1i 9	. . . . . 6 |- ( j e. I -> 2 e. ZZ )
9	6, 8	zmulcld 9608	. . . . 5 |- ( j e. I -> ( j x. 2 ) e. ZZ )
10		id 19	. . . . . 6 |- ( j e. I -> j e. I )
11		oveq1 6025	. . . . . . . 8 |- ( i = j -> ( i x. 2 ) = ( j x. 2 ) )
12	11	eqeq2d 2243	. . . . . . 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 2232	. . . . . 6 |- ( j e. I -> ( j x. 2 ) = ( j x. 2 ) )
15	10, 13, 14	rspcedvd 2916	. . . . 5 |- ( j e. I -> E. i e. I ( j x. 2 ) = ( i x. 2 ) )
16	3, 9, 15	elrabd 2964	. . . 4 |- ( j e. I -> ( j x. 2 ) e. { x e. ZZ | E. i e. I x = ( i x. 2 ) } )
17	1, 16	fmpti 5799	. . 3 |- F : I --> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
18		oveq1 6025	. . . . . . 7 |- ( j = y -> ( j x. 2 ) = ( y x. 2 ) )
19		simpl 109	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. I )
20		elfzelz 10260	. . . . . . . . . 10 |- ( y e. ( A ... B ) -> y e. ZZ )
21	20, 5	eleq2s 2326	. . . . . . . . 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 9608	. . . . . . . . 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 5740	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` y ) = ( y x. 2 ) )
28		oveq1 6025	. . . . . . 7 |- ( j = z -> ( j x. 2 ) = ( z x. 2 ) )
29		simpr 110	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. I )
30		elfzelz 10260	. . . . . . . . . 10 |- ( z e. ( A ... B ) -> z e. ZZ )
31	30, 5	eleq2s 2326	. . . . . . . . 9 |- ( z e. I -> z e. ZZ )
32	7	a1i 9	. . . . . . . . 9 |- ( z e. I -> 2 e. ZZ )
33	31, 32	zmulcld 9608	. . . . . . . 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 5740	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` z ) = ( z x. 2 ) )
36	27, 35	eqeq12d 2246	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) <-> ( y x. 2 ) = ( z x. 2 ) ) )
37	21	zcnd 9603	. . . . . . . 8 |- ( y e. I -> y e. CC )
38	37	adantr 276	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. CC )
39	31	zcnd 9603	. . . . . . . 8 |- ( z e. I -> z e. CC )
40	39	adantl 277	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. CC )
41		2cnd 9216	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 e. CC )
42		2ap0 9236	. . . . . . . 8 |- 2 # 0
43	42	a1i 9	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 # 0 )
44	38, 40, 41, 43	mulcanap2d 8842	. . . . . 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 2618	. . 3 |- A. y e. I A. z e. I ( ( F ` y ) = ( F ` z ) -> y = z )
48		dff13 5909	. . 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 950	. 2 |- F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
50		oveq1 6025	. . . . . . 7 |- ( j = i -> ( j x. 2 ) = ( i x. 2 ) )
51	50	eqeq2d 2243	. . . . . 6 |- ( j = i -> ( x = ( j x. 2 ) <-> x = ( i x. 2 ) ) )
52	51	cbvrexvw 2772	. . . . 5 |- ( E. j e. I x = ( j x. 2 ) <-> E. i e. I x = ( i x. 2 ) )
53		elfzelz 10260	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> i e. ZZ )
54	7	a1i 9	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> 2 e. ZZ )
55	53, 54	zmulcld 9608	. . . . . . . . 9 |- ( i e. ( A ... B ) -> ( i x. 2 ) e. ZZ )
56	55, 5	eleq2s 2326	. . . . . . . 8 |- ( i e. I -> ( i x. 2 ) e. ZZ )
57		eleq1 2294	. . . . . . . 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 2644	. . . . . 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 2347	. . 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 4980	. . 3 |- ran F = { x | E. j e. I x = ( j x. 2 ) }
64		df-rab 2519	. . 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 2262	. 2 |- ran F = { x e. ZZ | E. i e. I x = ( i x. 2 ) }
66		dff1o5 5592	. 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 950	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 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   {crab 2514   class class class wbr 4088   |-> cmpt 4150   ran crn 4726   -->wf 5322   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6018   CCcc 8030   0cc0 8032   x. cmul 8037   # cap 8761   2c2 9194   ZZcz 9479   ...cfz 10243

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244

This theorem is referenced by:  2lgslem1  15823