HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem cru 7988
Description: The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130.
Assertion
Ref Expression
cru |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (_i x. B)) = (C + (_i x. D)) <-> (A = C /\ B = D)))
Proof of Theorem cru
StepHypRef Expression
1 opreq1 4889 . . . 4 |- (A = if(A e. RR, A, 0) -> (A + (_i x. B)) = (if(A e. RR, A, 0) + (_i x. B)))
21eqeq1d 1892 . . 3 |- (A = if(A e. RR, A, 0) -> ((A + (_i x. B)) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) + (_i x. B)) = (C + (_i x. D))))
3 eqeq1 1890 . . . 4 |- (A = if(A e. RR, A, 0) -> (A = C <-> if(A e. RR, A, 0) = C))
43anbi1d 679 . . 3 |- (A = if(A e. RR, A, 0) -> ((A = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ B = D)))
52, 4bibi12d 691 . 2 |- (A = if(A e. RR, A, 0) -> (((A + (_i x. B)) = (C + (_i x. D)) <-> (A = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (_i x. B)) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) = C /\ B = D))))
6 opreq2 4890 . . . . 5 |- (B = if(B e. RR, B, 0) -> (_i x. B) = (_i x. if(B e. RR, B, 0)))
76opreq2d 4898 . . . 4 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0) + (_i x. B)) = (if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))))
87eqeq1d 1892 . . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) + (_i x. B)) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (C + (_i x. D))))
9 eqeq1 1890 . . . 4 |- (B = if(B e. RR, B, 0) -> (B = D <-> if(B e. RR, B, 0) = D))
109anbi2d 678 . . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)))
118, 10bibi12d 691 . 2 |- (B = if(B e. RR, B, 0) -> (((if(A e. RR, A, 0) + (_i x. B)) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D))))
12 opreq1 4889 . . . 4 |- (C = if(C e. RR, C, 0) -> (C + (_i x. D)) = (if(C e. RR, C, 0) + (_i x. D)))
1312eqeq2d 1895 . . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. D))))
14 eqeq2 1893 . . . 4 |- (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)))
1514anbi1d 679 . . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)))
1613, 15bibi12d 691 . 2 |- (C = if(C e. RR, C, 0) -> (((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (C + (_i x. D)) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. D)) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D))))
17 opreq2 4890 . . . . 5 |- (D = if(D e. RR, D, 0) -> (_i x. D) = (_i x. if(D e. RR, D, 0)))
1817opreq2d 4898 . . . 4 |- (D = if(D e. RR, D, 0) -> (if(C e. RR, C, 0) + (_i x. D)) = (if(C e. RR, C, 0) + (_i x. if(D e. RR, D, 0))))
1918eqeq2d 1895 . . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. D)) <-> (if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. if(D e. RR, D, 0)))))
20 eqeq2 1893 . . . 4 |- (D = if(D e. RR, D, 0) -> (if(B e. RR, B, 0) = D <-> if(B e. RR, B, 0) = if(D e. RR, D, 0)))
2120anbi2d 678 . . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0))))
2219, 21bibi12d 691 . 2 |- (D = if(D e. RR, D, 0) -> (((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. D)) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. if(D e. RR, D, 0))) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))))
23 0re 6603 . . . 4 |- 0 e. RR
2423elimel 3025 . . 3 |- if(A e. RR, A, 0) e. RR
2523elimel 3025 . . 3 |- if(B e. RR, B, 0) e. RR
2623elimel 3025 . . 3 |- if(C e. RR, C, 0) e. RR
2723elimel 3025 . . 3 |- if(D e. RR, D, 0) e. RR
2824, 25, 26, 27crui 7987 . 2 |- ((if(A e. RR, A, 0) + (_i x. if(B e. RR, B, 0))) = (if(C e. RR, C, 0) + (_i x. if(D e. RR, D, 0))) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))
295, 11, 16, 22, 28dedth4h 3017 1 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (_i x. B)) = (C + (_i x. D)) <-> (A = C /\ B = D)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982  (class class class)co 4884  RRcr 6385  0cc0 6386  _ici 6388   + caddc 6389   x. cmul 6391
This theorem is referenced by:  creur 7992  creui 7993  rimul 7994  replim 8011  crre 8019  crim 8020  cj11OLD 8081  efieq 8715  sinperlem1 10035  efifolem7 10082  efif1lem3 10086  eff1i 10098
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-div 6892
Copyright terms: Public domain