Theorem cnvtr 15016

Description: Converse of a translation.

Assertion

Ref	Expression
cnvtr	|- (A e. RR -> `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)))

Distinct variable group:   x,A

Proof of Theorem cnvtr

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . . . . . . . . 11 |- ((x e. RR /\ y = (x + A)) -> A.u(x e. RR /\ y = (x + A)))
2		ax-17 1317	. . . . . . . . . . 11 |- ((x e. RR /\ y = (x + A)) -> A.v(x e. RR /\ y = (x + A)))
3		ax-17 1317	. . . . . . . . . . 11 |- ((u e. RR /\ v = (u + A)) -> A.x(u e. RR /\ v = (u + A)))
4		ax-17 1317	. . . . . . . . . . 11 |- ((u e. RR /\ v = (u + A)) -> A.y(u e. RR /\ v = (u + A)))
5		eleq1 1957	. . . . . . . . . . . . 13 |- (x = u -> (x e. RR <-> u e. RR))
6	5	adantr 425	. . . . . . . . . . . 12 |- ((x = u /\ y = v) -> (x e. RR <-> u e. RR))
7		eqeq1 1890	. . . . . . . . . . . . 13 |- (y = v -> (y = (x + A) <-> v = (x + A)))
8		opreq1 4889	. . . . . . . . . . . . . 14 |- (x = u -> (x + A) = (u + A))
9	8	eqeq2d 1895	. . . . . . . . . . . . 13 |- (x = u -> (v = (x + A) <-> v = (u + A)))
10	7, 9	sylan9bbr 600	. . . . . . . . . . . 12 |- ((x = u /\ y = v) -> (y = (x + A) <-> v = (u + A)))
11	6, 10	anbi12d 690	. . . . . . . . . . 11 |- ((x = u /\ y = v) -> ((x e. RR /\ y = (x + A)) <-> (u e. RR /\ v = (u + A))))
12	1, 2, 3, 4, 11	cbvopab 3403	. . . . . . . . . 10 |- {<.x, y>. | (x e. RR /\ y = (x + A))} = {<.u, v>. | (u e. RR /\ v = (u + A))}
13		df-mpt 5006	. . . . . . . . . 10 |- (x e. RR |-> (x + A)) = {<.x, y>. | (x e. RR /\ y = (x + A))}
14		df-mpt 5006	. . . . . . . . . 10 |- (u e. RR |-> (u + A)) = {<.u, v>. | (u e. RR /\ v = (u + A))}
15	12, 13, 14	3eqtr4i 1921	. . . . . . . . 9 |- (x e. RR |-> (x + A)) = (u e. RR |-> (u + A))
16	15	trnij 15015	. . . . . . . 8 |- (A e. RR -> (x e. RR |-> (x + A)):RR-1-1-onto->RR)
17	16	adantr 425	. . . . . . 7 |- ((A e. RR /\ x e. RR) -> (x e. RR |-> (x + A)):RR-1-1-onto->RR)
18		resubcl 6601	. . . . . . . 8 |- ((x e. RR /\ A e. RR) -> (x - A) e. RR)
19	18	ancoms 484	. . . . . . 7 |- ((A e. RR /\ x e. RR) -> (x - A) e. RR)
20	17, 19	jca 310	. . . . . 6 |- ((A e. RR /\ x e. RR) -> ((x e. RR |-> (x + A)):RR-1-1-onto->RR /\ (x - A) e. RR))
21		eleq1 1957	. . . . . . . . . . . . 13 |- (y = x -> (y e. RR <-> x e. RR))
22	21	adantr 425	. . . . . . . . . . . 12 |- ((y = x /\ z = w) -> (y e. RR <-> x e. RR))
23		eqeq1 1890	. . . . . . . . . . . . 13 |- (z = w -> (z = (y + A) <-> w = (y + A)))
24		opreq1 4889	. . . . . . . . . . . . . 14 |- (y = x -> (y + A) = (x + A))
25	24	eqeq2d 1895	. . . . . . . . . . . . 13 |- (y = x -> (w = (y + A) <-> w = (x + A)))
26	23, 25	sylan9bbr 600	. . . . . . . . . . . 12 |- ((y = x /\ z = w) -> (z = (y + A) <-> w = (x + A)))
27	22, 26	anbi12d 690	. . . . . . . . . . 11 |- ((y = x /\ z = w) -> ((y e. RR /\ z = (y + A)) <-> (x e. RR /\ w = (x + A))))
28	27	cbvopabv 3404	. . . . . . . . . 10 |- {<.y, z>. | (y e. RR /\ z = (y + A))} = {<.x, w>. | (x e. RR /\ w = (x + A))}
29		df-mpt 5006	. . . . . . . . . 10 |- (y e. RR |-> (y + A)) = {<.y, z>. | (y e. RR /\ z = (y + A))}
30		df-mpt 5006	. . . . . . . . . 10 |- (x e. RR |-> (x + A)) = {<.x, w>. | (x e. RR /\ w = (x + A))}
31	28, 29, 30	3eqtr4ri 1923	. . . . . . . . 9 |- (x e. RR |-> (x + A)) = (y e. RR |-> (y + A))
32	31	a1i 8	. . . . . . . 8 |- ((A e. RR /\ x e. RR) -> (x e. RR |-> (x + A)) = (y e. RR |-> (y + A)))
33	32	fveq1d 4683	. . . . . . 7 |- ((A e. RR /\ x e. RR) -> ((x e. RR |-> (x + A))` (x - A)) = ((y e. RR |-> (y + A))` (x - A)))
34		readdcl 6455	. . . . . . . . . . 11 |- (((x - A) e. RR /\ A e. RR) -> ((x - A) + A) e. RR)
35	18, 34	sylancom 531	. . . . . . . . . 10 |- ((x e. RR /\ A e. RR) -> ((x - A) + A) e. RR)
36	18, 35	jca 310	. . . . . . . . 9 |- ((x e. RR /\ A e. RR) -> ((x - A) e. RR /\ ((x - A) + A) e. RR))
37	36	ancoms 484	. . . . . . . 8 |- ((A e. RR /\ x e. RR) -> ((x - A) e. RR /\ ((x - A) + A) e. RR))
38		opreq1 4889	. . . . . . . . . 10 |- (y = (x - A) -> (y + A) = ((x - A) + A))
39	38, 29	fvopab4g 4742	. . . . . . . . 9 |- (((x - A) e. RR /\ ((x - A) + A) e. RR) -> ((y e. RR |-> (y + A))` (x - A)) = ((x - A) + A))
40		npcan 6559	. . . . . . . . . . 11 |- ((x e. CC /\ A e. CC) -> ((x - A) + A) = x)
41		recn 6466	. . . . . . . . . . 11 |- (x e. RR -> x e. CC)
42		recn 6466	. . . . . . . . . . 11 |- (A e. RR -> A e. CC)
43	40, 41, 42	syl2an 503	. . . . . . . . . 10 |- ((x e. RR /\ A e. RR) -> ((x - A) + A) = x)
44	43	ancoms 484	. . . . . . . . 9 |- ((A e. RR /\ x e. RR) -> ((x - A) + A) = x)
45	39, 44	sylan9eqr 1951	. . . . . . . 8 |- (((A e. RR /\ x e. RR) /\ ((x - A) e. RR /\ ((x - A) + A) e. RR)) -> ((y e. RR |-> (y + A))` (x - A)) = x)
46	37, 45	mpdan 768	. . . . . . 7 |- ((A e. RR /\ x e. RR) -> ((y e. RR |-> (y + A))` (x - A)) = x)
47	33, 46	eqtrd 1925	. . . . . 6 |- ((A e. RR /\ x e. RR) -> ((x e. RR |-> (x + A))` (x - A)) = x)
48		f1ocnvfv 4856	. . . . . 6 |- (((x e. RR |-> (x + A)):RR-1-1-onto->RR /\ (x - A) e. RR) -> (((x e. RR |-> (x + A))` (x - A)) = x -> (`'(x e. RR |-> (x + A))` x) = (x - A)))
49	20, 47, 48	sylc 83	. . . . 5 |- ((A e. RR /\ x e. RR) -> (`'(x e. RR |-> (x + A))` x) = (x - A))
50		simpr 350	. . . . . 6 |- ((A e. RR /\ x e. RR) -> x e. RR)
51		eqid 1884	. . . . . . 7 |- (x e. RR |-> (x - A)) = (x e. RR |-> (x - A))
52	51	fvopab2b 14476	. . . . . 6 |- ((x e. RR /\ (x - A) e. RR) -> ((x e. RR |-> (x - A))` x) = (x - A))
53	50, 19, 52	syl11anc 524	. . . . 5 |- ((A e. RR /\ x e. RR) -> ((x e. RR |-> (x - A))` x) = (x - A))
54	49, 53	eqtr4d 1928	. . . 4 |- ((A e. RR /\ x e. RR) -> (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))
55	54	r19.21aiva 2176	. . 3 |- (A e. RR -> A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))
56		eqid 1884	. . 3 |- RR = RR
57	55, 56	jctil 316	. 2 |- (A e. RR -> (RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x)))
58		dff1o4 4644	. . . . 5 |- ((x e. RR |-> (x + A)):RR-1-1-onto->RR <-> ((x e. RR |-> (x + A)) Fn RR /\ `'(x e. RR |-> (x + A)) Fn RR))
59	16, 58	sylib 215	. . . 4 |- (A e. RR -> ((x e. RR |-> (x + A)) Fn RR /\ `'(x e. RR |-> (x + A)) Fn RR))
60	59	simprd 352	. . 3 |- (A e. RR -> `'(x e. RR |-> (x + A)) Fn RR)
61		elisset 2299	. . . . . . 7 |- ((x - A) e. RR -> (x - A) e. _V)
62	18, 61	syl 12	. . . . . 6 |- ((x e. RR /\ A e. RR) -> (x - A) e. _V)
63	62	ancoms 484	. . . . 5 |- ((A e. RR /\ x e. RR) -> (x - A) e. _V)
64	63	r19.21aiva 2176	. . . 4 |- (A e. RR -> A.x e. RR (x - A) e. _V)
65	51	fopab2ga 14478	. . . 4 |- (A.x e. RR (x - A) e. _V <-> (x e. RR |-> (x - A)) Fn RR)
66	64, 65	sylib 215	. . 3 |- (A e. RR -> (x e. RR |-> (x - A)) Fn RR)
67		eqidd 1885	. . . . . 6 |- (((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) /\ `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))) -> RR = RR)
68		hbmpt1 5010	. . . . . . . . 9 |- (w e. (x e. RR |-> (x + A)) -> A.x w e. (x e. RR |-> (x + A)))
69	68	hbcnv 4139	. . . . . . . 8 |- (w e. `'(x e. RR |-> (x + A)) -> A.x w e. `'(x e. RR |-> (x + A)))
70		hbmpt1 5010	. . . . . . . 8 |- (w e. (x e. RR |-> (x - A)) -> A.x w e. (x e. RR |-> (x - A)))
71	69, 70	eqfnfv2f 4770	. . . . . . 7 |- ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> (`'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)) <-> A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x)))
72	71	biimpa 460	. . . . . 6 |- (((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) /\ `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))) -> A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))
73	67, 72	jca 310	. . . . 5 |- (((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) /\ `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))) -> (RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x)))
74	73	ex 402	. . . 4 |- ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> (`'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)) -> (RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))))
75	71	biimprcd 173	. . . . . 6 |- (A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x) -> ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))))
76	75	adantl 424	. . . . 5 |- ((RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x)) -> ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))))
77	76	com12 14	. . . 4 |- ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> ((RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x)) -> `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A))))
78	74, 77	impbid 574	. . 3 |- ((`'(x e. RR |-> (x + A)) Fn RR /\ (x e. RR |-> (x - A)) Fn RR) -> (`'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)) <-> (RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))))
79	60, 66, 78	syl11anc 524	. 2 |- (A e. RR -> (`'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)) <-> (RR = RR /\ A.x e. RR (`'(x e. RR |-> (x + A))` x) = ((x e. RR |-> (x - A))` x))))
80	57, 79	mpbird 213	1 |- (A e. RR -> `'(x e. RR |-> (x + A)) = (x e. RR |-> (x - A)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {copab 3395  `'ccnv 3985   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  CCcc 6384  RRcr 6385   + caddc 6389   - cmin 6445

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-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-sub 6511  df-neg 6513

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