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Theorem cnegexlem3 6965
Description: Existence of real number difference. Lemma for cnegex 6966. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem3  b  RR  RR  c  RR  b  +  c
Distinct variable group:    b, c,

Proof of Theorem cnegexlem3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 readdcl 6785 . . . . . 6  b  RR  RR  b  +  RR
2 ax-rnegex 6772 . . . . . 6  b  +  RR  c  RR  b  +  +  c  0
31, 2syl 14 . . . . 5  b  RR  RR  c  RR  b  +  +  c  0
43adantlr 446 . . . 4  b  RR  RR  RR  c  RR  b  +  +  c  0
54adantr 261 . . 3  b  RR  RR  RR  +  0  c  RR  b  +  +  c  0
6 recn 6792 . . . . . . . 8  b  RR  b  CC
7 recn 6792 . . . . . . . 8  RR  CC
86, 7anim12i 321 . . . . . . 7  b  RR  RR  b  CC  CC
98anim1i 323 . . . . . 6  b  RR  RR  RR  b  CC  CC  RR
109anim1i 323 . . . . 5  b  RR  RR  RR  +  0  b  CC  CC  RR  +  0
11 recn 6792 . . . . 5  c  RR  c  CC
12 recn 6792 . . . . . . . . . 10  RR  CC
13 add32 6947 . . . . . . . . . . . 12  b  CC  CC  c  CC  b  +  +  c  b  +  c  +
14133expa 1103 . . . . . . . . . . 11  b  CC  CC  c  CC  b  +  +  c  b  +  c  +
15 addcl 6784 . . . . . . . . . . . . 13  b  CC  c  CC  b  +  c  CC
16 addcom 6927 . . . . . . . . . . . . 13  b  +  c  CC  CC  b  +  c  +  +  b  +  c
1715, 16sylan 267 . . . . . . . . . . . 12  b  CC  c  CC  CC  b  +  c  +  +  b  +  c
1817an32s 502 . . . . . . . . . . 11  b  CC  CC  c  CC  b  +  c  +  +  b  +  c
1914, 18eqtr2d 2070 . . . . . . . . . 10  b  CC  CC  c  CC  +  b  +  c  b  +  +  c
2012, 19sylanl2 383 . . . . . . . . 9  b  CC  RR  c  CC  +  b  +  c  b  +  +  c
2120adantllr 450 . . . . . . . 8  b  CC  CC  RR  c  CC  +  b  +  c  b  +  +  c
2221adantlr 446 . . . . . . 7  b  CC  CC  RR  +  0  c  CC  +  b  +  c  b  +  +  c
23 addcom 6927 . . . . . . . . . . . 12  CC  CC  +  +
2423ancoms 255 . . . . . . . . . . 11  CC  CC  +  +
2512, 24sylan2 270 . . . . . . . . . 10  CC  RR  +  +
26 id 19 . . . . . . . . . 10  +  0  +  0
2725, 26sylan9eq 2089 . . . . . . . . 9  CC  RR  +  0  +  0
2827adantlll 449 . . . . . . . 8  b  CC  CC  RR  +  0  +  0
2928adantr 261 . . . . . . 7  b  CC  CC  RR  +  0  c  CC  +  0
3022, 29eqeq12d 2051 . . . . . 6  b  CC  CC  RR  +  0  c  CC  +  b  +  c  +  b  +  +  c  0
31 simplr 482 . . . . . . . 8  b  CC  CC  RR  c  CC  RR
3215adantlr 446 . . . . . . . . 9  b  CC  CC  c  CC  b  +  c  CC
3332adantlr 446 . . . . . . . 8  b  CC  CC  RR  c  CC 
b  +  c  CC
34 simpllr 486 . . . . . . . 8  b  CC  CC  RR  c  CC  CC
35 cnegexlem1 6963 . . . . . . . 8  RR  b  +  c  CC  CC  + 
b  +  c  +  b  +  c
3631, 33, 34, 35syl3anc 1134 . . . . . . 7  b  CC  CC  RR  c  CC  + 
b  +  c  +  b  +  c
3736adantlr 446 . . . . . 6  b  CC  CC  RR  +  0  c  CC  +  b  +  c  +  b  +  c
3830, 37bitr3d 179 . . . . 5  b  CC  CC  RR  +  0  c  CC  b  +  +  c  0  b  +  c
3910, 11, 38syl2an 273 . . . 4  b  RR  RR  RR  +  0  c  RR  b  +  +  c  0  b  +  c
4039rexbidva 2317 . . 3  b  RR  RR  RR  +  0  c  RR  b  +  +  c  0  c  RR  b  +  c
415, 40mpbid 135 . 2  b  RR  RR  RR  +  0  c  RR  b  +  c
42 ax-rnegex 6772 . . 3  RR  RR  +  0
4342adantl 262 . 2  b  RR  RR  RR  +  0
4441, 43r19.29a 2448 1  b  RR  RR  c  RR  b  +  c
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455   CCcc 6689   RRcr 6690   0cc0 6691    + caddc 6694
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 6755  ax-1cn 6756  ax-icn 6758  ax-addcl 6759  ax-addrcl 6760  ax-mulcl 6761  ax-addcom 6763  ax-addass 6765  ax-i2m1 6768  ax-0id 6771  ax-rnegex 6772
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:  cnegex  6966
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