Theorem ismrer1 16024

Description: An isometry between RR and RR^1.

Hypotheses

Ref	Expression
ismrer1.1	|- R = ((abs o. - ) |` (RR X. RR))
ismrer1.2	|- F = {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}

Assertion

Ref	Expression
ismrer1	|- F e. (RIsmty(RRn` 1))

Distinct variable group:   x,R,y

Proof of Theorem ismrer1

Step	Hyp	Ref	Expression
1		ismrer1.1	. . . 4 |- R = ((abs o. - ) |` (RR X. RR))
2	1	remet 9188	. . 3 |- R e. Met
3		1nn 7117	. . . 4 |- 1 e. NN
4		rrnmet 16016	. . . 4 |- (1 e. NN -> (RRn` 1) e. Met)
5	3, 4	ax-mp 7	. . 3 |- (RRn` 1) e. Met
6	1	remetba 9187	. . . 4 |- RR = dom dom R
7		rrndm 16015	. . . . . 6 |- (1 e. NN -> dom dom (RRn` 1) = (RR ^m (1...1)))
8	3, 7	ax-mp 7	. . . . 5 |- dom dom (RRn` 1) = (RR ^m (1...1))
9		1z 7368	. . . . . . 7 |- 1 e. ZZ
10		fzsn 7684	. . . . . . 7 |- (1 e. ZZ -> (1...1) = {1})
11	9, 10	ax-mp 7	. . . . . 6 |- (1...1) = {1}
12	11	opreq2i 4893	. . . . 5 |- (RR ^m (1...1)) = (RR ^m {1})
13	8, 12	eqtr2i 1909	. . . 4 |- (RR ^m {1}) = dom dom (RRn` 1)
14	6, 13	isismty 15948	. . 3 |- ((R e. Met /\ (RRn` 1) e. Met) -> (F e. (RIsmty(RRn` 1)) <-> (F:RR-1-1-onto->(RR ^m {1}) /\ A.u e. RR A.v e. RR (uRv) = ((F` u)(RRn` 1)(F` v)))))
15	2, 5, 14	mp2an 761	. 2 |- (F e. (RIsmty(RRn` 1)) <-> (F:RR-1-1-onto->(RR ^m {1}) /\ A.u e. RR A.v e. RR (uRv) = ((F` u)(RRn` 1)(F` v))))
16		dff1o6 4853	. . 3 |- (F:RR-1-1-onto->(RR ^m {1}) <-> (F Fn RR /\ ran F = (RR ^m {1}) /\ A.u e. RR A.v e. RR ((F` u) = (F` v) -> u = v)))
17		snex 3492	. . . . 5 |- {1} e. _V
18		snex 3492	. . . . 5 |- {x} e. _V
19	17, 18	xpex 4096	. . . 4 |- ({1} X. {x}) e. _V
20		ismrer1.2	. . . 4 |- F = {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}
21	19, 20	fnopab2 4549	. . 3 |- F Fn RR
22		ax1cn 6422	. . . . . . . . 9 |- 1 e. CC
23	22	elisseti 2301	. . . . . . . 8 |- 1 e. _V
24		visset 2295	. . . . . . . 8 |- x e. _V
25	23, 24	xpsn 4808	. . . . . . 7 |- ({1} X. {x}) = {<.1, x>.}
26	25	eqeq2i 1894	. . . . . 6 |- (y = ({1} X. {x}) <-> y = {<.1, x>.})
27	26	rexbii 2128	. . . . 5 |- (E.x e. RR y = ({1} X. {x}) <-> E.x e. RR y = {<.1, x>.})
28	27	abbii 2006	. . . 4 |- {y | E.x e. RR y = ({1} X. {x})} = {y | E.x e. RR y = {<.1, x>.}}
29	20	rneqi 4187	. . . . 5 |- ran F = ran {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))}
30		rnopab2 4202	. . . . 5 |- ran {<.x, y>. | (x e. RR /\ y = ({1} X. {x}))} = {y | E.x e. RR y = ({1} X. {x})}
31	29, 30	eqtri 1908	. . . 4 |- ran F = {y | E.x e. RR y = ({1} X. {x})}
32		reex 6465	. . . . 5 |- RR e. _V
33	32, 23	mapsn 5404	. . . 4 |- (RR ^m {1}) = {y | E.x e. RR y = {<.1, x>.}}
34	28, 31, 33	3eqtr4i 1921	. . 3 |- ran F = (RR ^m {1})
35		sneq 3054	. . . . . . . 8 |- (x = u -> {x} = {u})
36		xpeq2 4017	. . . . . . . 8 |- ({x} = {u} -> ({1} X. {x}) = ({1} X. {u}))
37	35, 36	syl 12	. . . . . . 7 |- (x = u -> ({1} X. {x}) = ({1} X. {u}))
38		snex 3492	. . . . . . . 8 |- {u} e. _V
39	17, 38	xpex 4096	. . . . . . 7 |- ({1} X. {u}) e. _V
40	37, 20, 39	fvopab4 4743	. . . . . 6 |- (u e. RR -> (F` u) = ({1} X. {u}))
41		sneq 3054	. . . . . . . 8 |- (x = v -> {x} = {v})
42		xpeq2 4017	. . . . . . . 8 |- ({x} = {v} -> ({1} X. {x}) = ({1} X. {v}))
43	41, 42	syl 12	. . . . . . 7 |- (x = v -> ({1} X. {x}) = ({1} X. {v}))
44		snex 3492	. . . . . . . 8 |- {v} e. _V
45	17, 44	xpex 4096	. . . . . . 7 |- ({1} X. {v}) e. _V
46	43, 20, 45	fvopab4 4743	. . . . . 6 |- (v e. RR -> (F` v) = ({1} X. {v}))
47	40, 46	eqeqan12d 1901	. . . . 5 |- ((u e. RR /\ v e. RR) -> ((F` u) = (F` v) <-> ({1} X. {u}) = ({1} X. {v})))
48	23	snnz 3119	. . . . . . 7 |- {1} =/= (/)
49		visset 2295	. . . . . . . 8 |- u e. _V
50	49	snnz 3119	. . . . . . 7 |- {u} =/= (/)
51		xp11 4347	. . . . . . 7 |- (({1} =/= (/) /\ {u} =/= (/)) -> (({1} X. {u}) = ({1} X. {v}) <-> ({1} = {1} /\ {u} = {v})))
52	48, 50, 51	mp2an 761	. . . . . 6 |- (({1} X. {u}) = ({1} X. {v}) <-> ({1} = {1} /\ {u} = {v}))
53	49	sneqr 3147	. . . . . . 7 |- ({u} = {v} -> u = v)
54	53	adantl 424	. . . . . 6 |- (({1} = {1} /\ {u} = {v}) -> u = v)
55	52, 54	sylbi 216	. . . . 5 |- (({1} X. {u}) = ({1} X. {v}) -> u = v)
56	47, 55	syl6bi 231	. . . 4 |- ((u e. RR /\ v e. RR) -> ((F` u) = (F` v) -> u = v))
57	56	rgen2a 2160	. . 3 |- A.u e. RR A.v e. RR ((F` u) = (F` v) -> u = v)
58	16, 21, 34, 57	mpbir3an 1052	. 2 |- F:RR-1-1-onto->(RR ^m {1})
59		oprex 4907	. . . . . . . 8 |- ((((F` u)` 1) - ((F` v)` 1))^2) e. _V
60		fveq2 4681	. . . . . . . . . . 11 |- (k = 1 -> ((F` u)` k) = ((F` u)` 1))
61		fveq2 4681	. . . . . . . . . . 11 |- (k = 1 -> ((F` v)` k) = ((F` v)` 1))
62	60, 61	opreq12d 4900	. . . . . . . . . 10 |- (k = 1 -> (((F` u)` k) - ((F` v)` k)) = (((F` u)` 1) - ((F` v)` 1)))
63	62	opreq1d 4897	. . . . . . . . 9 |- (k = 1 -> ((((F` u)` k) - ((F` v)` k))^2) = ((((F` u)` 1) - ((F` v)` 1))^2))
64	63	fsum1i 8265	. . . . . . . 8 |- ((((((F` u)` 1) - ((F` v)` 1))^2) e. _V /\ 1 e. ZZ) -> sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2) = ((((F` u)` 1) - ((F` v)` 1))^2))
65	59, 9, 64	mp2an 761	. . . . . . 7 |- sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2) = ((((F` u)` 1) - ((F` v)` 1))^2)
66	65	fveq2i 4684	. . . . . 6 |- (sqr` sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2)) = (sqr` ((((F` u)` 1) - ((F` v)` 1))^2))
67	66	a1i 8	. . . . 5 |- ((u e. RR /\ v e. RR) -> (sqr` sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2)) = (sqr` ((((F` u)` 1) - ((F` v)` 1))^2)))
68	40	fveq1d 4683	. . . . . . . . 9 |- (u e. RR -> ((F` u)` 1) = (({1} X. {u})` 1))
69	23	snid 3069	. . . . . . . . . 10 |- 1 e. {1}
70	49	fvconst2 4822	. . . . . . . . . 10 |- (1 e. {1} -> (({1} X. {u})` 1) = u)
71	69, 70	ax-mp 7	. . . . . . . . 9 |- (({1} X. {u})` 1) = u
72	68, 71	syl6eq 1944	. . . . . . . 8 |- (u e. RR -> ((F` u)` 1) = u)
73	46	fveq1d 4683	. . . . . . . . 9 |- (v e. RR -> ((F` v)` 1) = (({1} X. {v})` 1))
74		visset 2295	. . . . . . . . . . 11 |- v e. _V
75	74	fvconst2 4822	. . . . . . . . . 10 |- (1 e. {1} -> (({1} X. {v})` 1) = v)
76	69, 75	ax-mp 7	. . . . . . . . 9 |- (({1} X. {v})` 1) = v
77	73, 76	syl6eq 1944	. . . . . . . 8 |- (v e. RR -> ((F` v)` 1) = v)
78	72, 77	opreqan12d 4902	. . . . . . 7 |- ((u e. RR /\ v e. RR) -> (((F` u)` 1) - ((F` v)` 1)) = (u - v))
79	78	opreq1d 4897	. . . . . 6 |- ((u e. RR /\ v e. RR) -> ((((F` u)` 1) - ((F` v)` 1))^2) = ((u - v)^2))
80	79	fveq2d 4685	. . . . 5 |- ((u e. RR /\ v e. RR) -> (sqr` ((((F` u)` 1) - ((F` v)` 1))^2)) = (sqr` ((u - v)^2)))
81		resubcl 6601	. . . . . 6 |- ((u e. RR /\ v e. RR) -> (u - v) e. RR)
82		absresq 8118	. . . . . . . . 9 |- ((u - v) e. RR -> ((abs` (u - v))^2) = ((u - v)^2))
83	82	eqcomd 1889	. . . . . . . 8 |- ((u - v) e. RR -> ((u - v)^2) = ((abs` (u - v))^2))
84	83	fveq2d 4685	. . . . . . 7 |- ((u - v) e. RR -> (sqr` ((u - v)^2)) = (sqr` ((abs` (u - v))^2)))
85		recn 6466	. . . . . . . . 9 |- ((u - v) e. RR -> (u - v) e. CC)
86		abscl 8084	. . . . . . . . 9 |- ((u - v) e. CC -> (abs` (u - v)) e. RR)
87	85, 86	syl 12	. . . . . . . 8 |- ((u - v) e. RR -> (abs` (u - v)) e. RR)
88		absge0 8105	. . . . . . . . 9 |- ((u - v) e. CC -> 0 <_ (abs` (u - v)))
89	85, 88	syl 12	. . . . . . . 8 |- ((u - v) e. RR -> 0 <_ (abs` (u - v)))
90		sqrsq 7972	. . . . . . . 8 |- (((abs` (u - v)) e. RR /\ 0 <_ (abs` (u - v))) -> (sqr` ((abs` (u - v))^2)) = (abs` (u - v)))
91	87, 89, 90	syl11anc 524	. . . . . . 7 |- ((u - v) e. RR -> (sqr` ((abs` (u - v))^2)) = (abs` (u - v)))
92	84, 91	eqtrd 1925	. . . . . 6 |- ((u - v) e. RR -> (sqr` ((u - v)^2)) = (abs` (u - v)))
93	81, 92	syl 12	. . . . 5 |- ((u e. RR /\ v e. RR) -> (sqr` ((u - v)^2)) = (abs` (u - v)))
94	67, 80, 93	3eqtrrd 1930	. . . 4 |- ((u e. RR /\ v e. RR) -> (abs` (u - v)) = (sqr` sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2)))
95	1	remetdval 9186	. . . 4 |- ((u e. RR /\ v e. RR) -> (uRv) = (abs` (u - v)))
96	3	a1i 8	. . . . 5 |- ((u e. RR /\ v e. RR) -> 1 e. NN)
97		f1of 4635	. . . . . . . . 9 |- (F:RR-1-1-onto->(RR ^m {1}) -> F:RR-->(RR ^m {1}))
98	58, 97	ax-mp 7	. . . . . . . 8 |- F:RR-->(RR ^m {1})
99	98	ffvelrni 4788	. . . . . . 7 |- (u e. RR -> (F` u) e. (RR ^m {1}))
100	99, 12	syl6eleqr 1982	. . . . . 6 |- (u e. RR -> (F` u) e. (RR ^m (1...1)))
101	100	adantr 425	. . . . 5 |- ((u e. RR /\ v e. RR) -> (F` u) e. (RR ^m (1...1)))
102	98	ffvelrni 4788	. . . . . . 7 |- (v e. RR -> (F` v) e. (RR ^m {1}))
103	102, 12	syl6eleqr 1982	. . . . . 6 |- (v e. RR -> (F` v) e. (RR ^m (1...1)))
104	103	adantl 424	. . . . 5 |- ((u e. RR /\ v e. RR) -> (F` v) e. (RR ^m (1...1)))
105		rrnmval 16014	. . . . 5 |- ((1 e. NN /\ (F` u) e. (RR ^m (1...1)) /\ (F` v) e. (RR ^m (1...1))) -> ((F` u)(RRn` 1)(F` v)) = (sqr` sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2)))
106	96, 101, 104, 105	syl111anc 1100	. . . 4 |- ((u e. RR /\ v e. RR) -> ((F` u)(RRn` 1)(F` v)) = (sqr` sum_k e. (1...1)((((F` u)` k) - ((F` v)` k))^2)))
107	94, 95, 106	3eqtr4d 1937	. . 3 |- ((u e. RR /\ v e. RR) -> (uRv) = ((F` u)(RRn` 1)(F` v)))
108	107	rgen2a 2160	. 2 |- A.u e. RR A.v e. RR (uRv) = ((F` u)(RRn` 1)(F` v))
109	15, 58, 108	mpbir2an 800	1 |- F e. (RIsmty(RRn` 1))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884   ^m cmap 5381  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   - cmin 6445   <_ cle 6448  NNcn 6449  ZZcz 6451  2c2 7145  ...cfz 7637  ^cexp 7811  sqrcsqr 7919  abscabs 8000  sum_csu 8239  Metcme 9066  Ismtycismty 15945  RRncrrn 16011

This theorem is referenced by:  reheibor 16025

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  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-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240  df-met 9070  df-ismty 15946  df-rrn 16012

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