Theorem 2lgslem1b 25117

Description: Lemma 2 for 2lgslem1 25119. (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		elfzelz 12342	. . . . . . 7 |- ( j e. ( A ... B ) -> j e. ZZ )
3		2lgslem1b.i	. . . . . . 7 |- I = ( A ... B )
4	2, 3	eleq2s 2719	. . . . . 6 |- ( j e. I -> j e. ZZ )
5		2z 11409	. . . . . . 7 |- 2 e. ZZ
6	5	a1i 11	. . . . . 6 |- ( j e. I -> 2 e. ZZ )
7	4, 6	zmulcld 11488	. . . . 5 |- ( j e. I -> ( j x. 2 ) e. ZZ )
8		id 22	. . . . . 6 |- ( j e. I -> j e. I )
9		oveq1 6657	. . . . . . . 8 |- ( i = j -> ( i x. 2 ) = ( j x. 2 ) )
10	9	eqeq2d 2632	. . . . . . 7 |- ( i = j -> ( ( j x. 2 ) = ( i x. 2 ) <-> ( j x. 2 ) = ( j x. 2 ) ) )
11	10	adantl 482	. . . . . 6 |- ( ( j e. I /\ i = j ) -> ( ( j x. 2 ) = ( i x. 2 ) <-> ( j x. 2 ) = ( j x. 2 ) ) )
12		eqidd 2623	. . . . . 6 |- ( j e. I -> ( j x. 2 ) = ( j x. 2 ) )
13	8, 11, 12	rspcedvd 3317	. . . . 5 |- ( j e. I -> E. i e. I ( j x. 2 ) = ( i x. 2 ) )
14		eqeq1 2626	. . . . . . 7 |- ( x = ( j x. 2 ) -> ( x = ( i x. 2 ) <-> ( j x. 2 ) = ( i x. 2 ) ) )
15	14	rexbidv 3052	. . . . . 6 |- ( x = ( j x. 2 ) -> ( E. i e. I x = ( i x. 2 ) <-> E. i e. I ( j x. 2 ) = ( i x. 2 ) ) )
16	15	elrab 3363	. . . . 5 |- ( ( j x. 2 ) e. { x e. ZZ | E. i e. I x = ( i x. 2 ) } <-> ( ( j x. 2 ) e. ZZ /\ E. i e. I ( j x. 2 ) = ( i x. 2 ) ) )
17	7, 13, 16	sylanbrc 698	. . . 4 |- ( j e. I -> ( j x. 2 ) e. { x e. ZZ | E. i e. I x = ( i x. 2 ) } )
18	1, 17	fmpti 6383	. . 3 |- F : I --> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
19	1	a1i 11	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> F = ( j e. I |-> ( j x. 2 ) ) )
20		oveq1 6657	. . . . . . . 8 |- ( j = y -> ( j x. 2 ) = ( y x. 2 ) )
21	20	adantl 482	. . . . . . 7 |- ( ( ( y e. I /\ z e. I ) /\ j = y ) -> ( j x. 2 ) = ( y x. 2 ) )
22		simpl 473	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. I )
23		ovexd 6680	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> ( y x. 2 ) e. _V )
24	19, 21, 22, 23	fvmptd 6288	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` y ) = ( y x. 2 ) )
25		oveq1 6657	. . . . . . . 8 |- ( j = z -> ( j x. 2 ) = ( z x. 2 ) )
26	25	adantl 482	. . . . . . 7 |- ( ( ( y e. I /\ z e. I ) /\ j = z ) -> ( j x. 2 ) = ( z x. 2 ) )
27		simpr 477	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. I )
28		ovexd 6680	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> ( z x. 2 ) e. _V )
29	19, 26, 27, 28	fvmptd 6288	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( F ` z ) = ( z x. 2 ) )
30	24, 29	eqeq12d 2637	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) <-> ( y x. 2 ) = ( z x. 2 ) ) )
31		elfzelz 12342	. . . . . . . . . 10 |- ( y e. ( A ... B ) -> y e. ZZ )
32	31, 3	eleq2s 2719	. . . . . . . . 9 |- ( y e. I -> y e. ZZ )
33	32	zcnd 11483	. . . . . . . 8 |- ( y e. I -> y e. CC )
34	33	adantr 481	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> y e. CC )
35		elfzelz 12342	. . . . . . . . . 10 |- ( z e. ( A ... B ) -> z e. ZZ )
36	35, 3	eleq2s 2719	. . . . . . . . 9 |- ( z e. I -> z e. ZZ )
37	36	zcnd 11483	. . . . . . . 8 |- ( z e. I -> z e. CC )
38	37	adantl 482	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> z e. CC )
39		2cnd 11093	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 e. CC )
40		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
41	40	a1i 11	. . . . . . 7 |- ( ( y e. I /\ z e. I ) -> 2 =/= 0 )
42	34, 38, 39, 41	mulcan2d 10661	. . . . . 6 |- ( ( y e. I /\ z e. I ) -> ( ( y x. 2 ) = ( z x. 2 ) <-> y = z ) )
43	42	biimpd 219	. . . . 5 |- ( ( y e. I /\ z e. I ) -> ( ( y x. 2 ) = ( z x. 2 ) -> y = z ) )
44	30, 43	sylbid 230	. . . 4 |- ( ( y e. I /\ z e. I ) -> ( ( F ` y ) = ( F ` z ) -> y = z ) )
45	44	rgen2 2975	. . 3 |- A. y e. I A. z e. I ( ( F ` y ) = ( F ` z ) -> y = z )
46		dff13 6512	. . 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 ) ) )
47	18, 45, 46	mpbir2an 955	. 2 |- F : I -1-1-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
48		oveq1 6657	. . . . . . 7 |- ( j = i -> ( j x. 2 ) = ( i x. 2 ) )
49	48	eqeq2d 2632	. . . . . 6 |- ( j = i -> ( x = ( j x. 2 ) <-> x = ( i x. 2 ) ) )
50	49	cbvrexv 3172	. . . . 5 |- ( E. j e. I x = ( j x. 2 ) <-> E. i e. I x = ( i x. 2 ) )
51		elfzelz 12342	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> i e. ZZ )
52	5	a1i 11	. . . . . . . . . 10 |- ( i e. ( A ... B ) -> 2 e. ZZ )
53	51, 52	zmulcld 11488	. . . . . . . . 9 |- ( i e. ( A ... B ) -> ( i x. 2 ) e. ZZ )
54	53, 3	eleq2s 2719	. . . . . . . 8 |- ( i e. I -> ( i x. 2 ) e. ZZ )
55		eleq1 2689	. . . . . . . 8 |- ( x = ( i x. 2 ) -> ( x e. ZZ <-> ( i x. 2 ) e. ZZ ) )
56	54, 55	syl5ibrcom 237	. . . . . . 7 |- ( i e. I -> ( x = ( i x. 2 ) -> x e. ZZ ) )
57	56	rexlimiv 3027	. . . . . 6 |- ( E. i e. I x = ( i x. 2 ) -> x e. ZZ )
58	57	pm4.71ri 665	. . . . 5 |- ( E. i e. I x = ( i x. 2 ) <-> ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) )
59	50, 58	bitri 264	. . . 4 |- ( E. j e. I x = ( j x. 2 ) <-> ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) )
60	59	abbii 2739	. . 3 |- { x | E. j e. I x = ( j x. 2 ) } = { x | ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) }
61	1	rnmpt 5371	. . 3 |- ran F = { x | E. j e. I x = ( j x. 2 ) }
62		df-rab 2921	. . 3 |- { x e. ZZ | E. i e. I x = ( i x. 2 ) } = { x | ( x e. ZZ /\ E. i e. I x = ( i x. 2 ) ) }
63	60, 61, 62	3eqtr4i 2654	. 2 |- ran F = { x e. ZZ | E. i e. I x = ( i x. 2 ) }
64		dff1o5 6146	. 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 ) } ) )
65	47, 63, 64	mpbir2an 955	1 |- F : I -1-1-onto-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   |-> cmpt 4729   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   2c2 11070   ZZcz 11377   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  2lgslem1  25119