Theorem icoshftf1olem 7579

Description: Lemma for icoshftf1o 7580.

Hypotheses

Ref	Expression
icoshftf1o.1	|- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}
icoshftf1olem.1	|- G = {<.x, y>. | (x e. (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) /\ y = (x + if(C e. RR, C, 0)))}

Assertion

Ref	Expression
icoshftf1olem	|- ((A e. RR /\ B e. RR /\ C e. RR) -> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C)))

Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem icoshftf1olem

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . 5 |- (A = if(A e. RR, A, 0) -> (A[,)B) = (if(A e. RR, A, 0)[,)B))
2		f1oeq2 4631	. . . . 5 |- ((A[,)B) = (if(A e. RR, A, 0)[,)B) -> (G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((A + C)[,)(B + C))))
3	1, 2	syl 12	. . . 4 |- (A = if(A e. RR, A, 0) -> (G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((A + C)[,)(B + C))))
4		opreq1 4889	. . . . . 6 |- (A = if(A e. RR, A, 0) -> (A + C) = (if(A e. RR, A, 0) + C))
5	4	opreq1d 4897	. . . . 5 |- (A = if(A e. RR, A, 0) -> ((A + C)[,)(B + C)) = ((if(A e. RR, A, 0) + C)[,)(B + C)))
6		f1oeq3 4632	. . . . 5 |- (((A + C)[,)(B + C)) = ((if(A e. RR, A, 0) + C)[,)(B + C)) -> (G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C))))
7	5, 6	syl 12	. . . 4 |- (A = if(A e. RR, A, 0) -> (G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C))))
8	3, 7	bitrd 587	. . 3 |- (A = if(A e. RR, A, 0) -> (G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C))))
9		opreq2 4890	. . . . 5 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0)[,)B) = (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)))
10		f1oeq2 4631	. . . . 5 |- ((if(A e. RR, A, 0)[,)B) = (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) -> (G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C))))
11	9, 10	syl 12	. . . 4 |- (B = if(B e. RR, B, 0) -> (G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C))))
12		opreq1 4889	. . . . . 6 |- (B = if(B e. RR, B, 0) -> (B + C) = (if(B e. RR, B, 0) + C))
13	12	opreq2d 4898	. . . . 5 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) + C)[,)(B + C)) = ((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)))
14		f1oeq3 4632	. . . . 5 |- (((if(A e. RR, A, 0) + C)[,)(B + C)) = ((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)) -> (G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C))))
15	13, 14	syl 12	. . . 4 |- (B = if(B e. RR, B, 0) -> (G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C))))
16	11, 15	bitrd 587	. . 3 |- (B = if(B e. RR, B, 0) -> (G:(if(A e. RR, A, 0)[,)B)-1-1-onto->((if(A e. RR, A, 0) + C)[,)(B + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C))))
17		opreq2 4890	. . . . 5 |- (C = if(C e. RR, C, 0) -> (if(A e. RR, A, 0) + C) = (if(A e. RR, A, 0) + if(C e. RR, C, 0)))
18		opreq2 4890	. . . . 5 |- (C = if(C e. RR, C, 0) -> (if(B e. RR, B, 0) + C) = (if(B e. RR, B, 0) + if(C e. RR, C, 0)))
19	17, 18	opreq12d 4900	. . . 4 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)) = ((if(A e. RR, A, 0) + if(C e. RR, C, 0))[,)(if(B e. RR, B, 0) + if(C e. RR, C, 0))))
20		f1oeq3 4632	. . . 4 |- (((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)) = ((if(A e. RR, A, 0) + if(C e. RR, C, 0))[,)(if(B e. RR, B, 0) + if(C e. RR, C, 0))) -> (G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + if(C e. RR, C, 0))[,)(if(B e. RR, B, 0) + if(C e. RR, C, 0)))))
21	19, 20	syl 12	. . 3 |- (C = if(C e. RR, C, 0) -> (G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + C)[,)(if(B e. RR, B, 0) + C)) <-> G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + if(C e. RR, C, 0))[,)(if(B e. RR, B, 0) + if(C e. RR, C, 0)))))
22		icoshftf1olem.1	. . . 4 |- G = {<.x, y>. | (x e. (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) /\ y = (x + if(C e. RR, C, 0)))}
23		0re 6603	. . . . 5 |- 0 e. RR
24	23	elimel 3025	. . . 4 |- if(A e. RR, A, 0) e. RR
25	23	elimel 3025	. . . 4 |- if(B e. RR, B, 0) e. RR
26	23	elimel 3025	. . . 4 |- if(C e. RR, C, 0) e. RR
27	22, 24, 25, 26	icoshftf1oii 7578	. . 3 |- G:(if(A e. RR, A, 0)[,)if(B e. RR, B, 0))-1-1-onto->((if(A e. RR, A, 0) + if(C e. RR, C, 0))[,)(if(B e. RR, B, 0) + if(C e. RR, C, 0)))
28	8, 16, 21, 27	dedth3h 3016	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)))
29		iftrue 2989	. . . . . . . . 9 |- (A e. RR -> if(A e. RR, A, 0) = A)
30	29	3ad2ant1 897	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR) -> if(A e. RR, A, 0) = A)
31		iftrue 2989	. . . . . . . . 9 |- (B e. RR -> if(B e. RR, B, 0) = B)
32	31	3ad2ant2 898	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR) -> if(B e. RR, B, 0) = B)
33	30, 32	opreq12d 4900	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) = (A[,)B))
34	33	eleq2d 1964	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x e. (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) <-> x e. (A[,)B)))
35		iftrue 2989	. . . . . . . . 9 |- (C e. RR -> if(C e. RR, C, 0) = C)
36	35	3ad2ant3 899	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ C e. RR) -> if(C e. RR, C, 0) = C)
37	36	opreq2d 4898	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (x + if(C e. RR, C, 0)) = (x + C))
38	37	eqeq2d 1895	. . . . . 6 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (y = (x + if(C e. RR, C, 0)) <-> y = (x + C)))
39	34, 38	anbi12d 690	. . . . 5 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((x e. (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) /\ y = (x + if(C e. RR, C, 0))) <-> (x e. (A[,)B) /\ y = (x + C))))
40	39	opabbidv 3401	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> {<.x, y>. | (x e. (if(A e. RR, A, 0)[,)if(B e. RR, B, 0)) /\ y = (x + if(C e. RR, C, 0)))} = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))})
41		icoshftf1o.1	. . . 4 |- F = {<.x, y>. | (x e. (A[,)B) /\ y = (x + C))}
42	40, 22, 41	3eqtr4g 1953	. . 3 |- ((A e. RR /\ B e. RR /\ C e. RR) -> G = F)
43		f1oeq1 4630	. . 3 |- (G = F -> (G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C))))
44	42, 43	syl 12	. 2 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (G:(A[,)B)-1-1-onto->((A + C)[,)(B + C)) <-> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C))))
45	28, 44	mpbid 212	1 |- ((A e. RR /\ B e. RR /\ C e. RR) -> F:(A[,)B)-1-1-onto->((A + C)[,)(B + C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ifcif 2982  {copab 3395  -1-1-onto->wf1o 3997  (class class class)co 4884  RRcr 6385  0cc0 6386   + caddc 6389  [,)cico 7526

This theorem is referenced by:  icoshftf1o 7580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-ico 7530

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