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Theorem cnegexlem1 6983
Description: Addition cancellation of a real number from two complex numbers. Lemma for cnegex 6986. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem1  RR  CC  C  CC  +  +  C  C

Proof of Theorem cnegexlem1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-rnegex 6792 . . . 4  RR  RR  +  0
213ad2ant1 924 . . 3  RR  CC  C  CC  RR  +  0
3 recn 6812 . . . 4  RR  CC
4 recn 6812 . . . . . . 7  RR  CC
5 oveq2 5463 . . . . . . . . . . 11  +  +  C  +  +  +  +  C
6 simpr 103 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  CC
7 simpll 481 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  CC
8 simplrl 487 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  CC
96, 7, 8addassd 6847 . . . . . . . . . . . 12  CC  CC  C  CC  CC  +  +  +  +
10 simplrr 488 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  C  CC
116, 7, 10addassd 6847 . . . . . . . . . . . 12  CC  CC  C  CC  CC  +  +  C  +  +  C
129, 11eqeq12d 2051 . . . . . . . . . . 11  CC  CC  C  CC  CC  +  +  +  +  C  +  +  +  +  C
135, 12syl5ibr 145 . . . . . . . . . 10  CC  CC  C  CC  CC  +  +  C  +  +  +  +  C
1413adantr 261 . . . . . . . . 9  CC  CC  C  CC  CC  +  0  +  +  C  +  +  +  +  C
15 addcom 6947 . . . . . . . . . . . . 13  CC  CC  +  +
1615eqeq1d 2045 . . . . . . . . . . . 12  CC  CC  +  0  +  0
1716adantlr 446 . . . . . . . . . . 11  CC  CC  C  CC  CC  +  0  +  0
18 oveq1 5462 . . . . . . . . . . . . . . 15  +  0  +  +  0  +
19 oveq1 5462 . . . . . . . . . . . . . . 15  +  0  +  +  C  0  +  C
2018, 19eqeq12d 2051 . . . . . . . . . . . . . 14  +  0  +  +  +  +  C  0  +  0  +  C
2120adantl 262 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  +  0  +  +  +  +  C  0  +  0  +  C
22 addid2 6949 . . . . . . . . . . . . . . . 16  CC 
0  +
23 addid2 6949 . . . . . . . . . . . . . . . 16  C  CC 
0  +  C  C
2422, 23eqeqan12d 2052 . . . . . . . . . . . . . . 15  CC  C  CC  0  +  0  +  C  C
2524adantl 262 . . . . . . . . . . . . . 14  CC  CC  C  CC  0  +  0  +  C  C
2625ad2antrr 457 . . . . . . . . . . . . 13  CC  CC  C  CC  CC  +  0  0  +  0  +  C  C
2721, 26bitrd 177 . . . . . . . . . . . 12  CC  CC  C  CC  CC  +  0  +  +  +  +  C  C
2827ex 108 . . . . . . . . . . 11  CC  CC  C  CC  CC  +  0  +  +  +  +  C  C
2917, 28sylbid 139 . . . . . . . . . 10  CC  CC  C  CC  CC  +  0  +  +  +  +  C  C
3029imp 115 . . . . . . . . 9  CC  CC  C  CC  CC  +  0  +  +  +  +  C  C
3114, 30sylibd 138 . . . . . . . 8  CC  CC  C  CC  CC  +  0  +  +  C  C
3231ex 108 . . . . . . 7  CC  CC  C  CC  CC  +  0  +  +  C  C
334, 32sylan2 270 . . . . . 6  CC  CC  C  CC  RR  +  0  +  +  C  C
3433rexlimdva 2427 . . . . 5  CC  CC  C  CC  RR  +  0  +  +  C  C
35343impb 1099 . . . 4  CC  CC  C  CC  RR  +  0  +  +  C  C
363, 35syl3an1 1167 . . 3  RR  CC  C  CC  RR  +  0  +  +  C  C
372, 36mpd 13 . 2  RR  CC  C  CC  +  +  C  C
38 oveq2 5463 . 2  C  +  +  C
3937, 38impbid1 130 1  RR  CC  C  CC  +  +  C  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455   CCcc 6709   RRcr 6710   0cc0 6711    + caddc 6714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  cnegexlem3  6985
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