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Theorem addid1 9671
Description:  0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addid1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )

Proof of Theorem addid1
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9506 . 2  |-  1  e.  RR
2 ax-rnegex 9474 . 2  |-  ( 1  e.  RR  ->  E. c  e.  RR  ( 1  +  c )  =  0 )
3 ax-1ne0 9472 . . . . . 6  |-  1  =/=  0
4 oveq2 6204 . . . . . . . . . 10  |-  ( c  =  0  ->  (
1  +  c )  =  ( 1  +  0 ) )
54eqeq1d 2384 . . . . . . . . 9  |-  ( c  =  0  ->  (
( 1  +  c )  =  0  <->  (
1  +  0 )  =  0 ) )
65biimpcd 224 . . . . . . . 8  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
( 1  +  0 )  =  0 ) )
7 oveq2 6204 . . . . . . . . 9  |-  ( ( 1  +  0 )  =  0  ->  (
( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
8 ax-icn 9462 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
98, 8mulcli 9512 . . . . . . . . . . . . . 14  |-  ( _i  x.  _i )  e.  CC
109, 9mulcli 9512 . . . . . . . . . . . . 13  |-  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC
11 ax-1cn 9461 . . . . . . . . . . . . 13  |-  1  e.  CC
12 0cn 9499 . . . . . . . . . . . . 13  |-  0  e.  CC
1310, 11, 12adddii 9517 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
1410mulid1i 9509 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  =  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )
15 mul01 9670 . . . . . . . . . . . . . . 15  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC  ->  ( (
( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0 )
1610, 15ax-mp 5 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0
17 ax-i2m1 9471 . . . . . . . . . . . . . 14  |-  ( ( _i  x.  _i )  +  1 )  =  0
1816, 17eqtr4i 2414 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  ( ( _i  x.  _i )  +  1 )
1914, 18oveq12i 6208 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2013, 19eqtri 2411 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2120, 16eqeq12i 2402 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )  =  0 )
2210, 9, 11addassi 9515 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )
239mulid1i 9509 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  1 )  =  ( _i  x.  _i )
2423oveq2i 6207 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )
259, 9, 11adddii 9517 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1 ) )
2617oveq2i 6207 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( _i  x.  _i )  x.  0
)
27 mul01 9670 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  x.  _i )  e.  CC  ->  (
( _i  x.  _i )  x.  0 )  =  0 )
289, 27ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  0 )  =  0
2926, 28eqtri 2411 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  0
3025, 29eqtr3i 2413 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  0
3124, 30eqtr3i 2413 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( _i  x.  _i ) )  =  0
3231oveq1i 6206 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( 0  +  1 )
3322, 32eqtr3i 2413 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1
) )  =  ( 0  +  1 )
34 00id 9666 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
3534eqcomi 2395 . . . . . . . . . . 11  |-  0  =  ( 0  +  0 )
3633, 35eqeq12i 2402 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )  =  0  <->  ( 0  +  1 )  =  ( 0  +  0 ) )
37 0re 9507 . . . . . . . . . . 11  |-  0  e.  RR
38 readdcan 9665 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
391, 37, 37, 38mp3an 1322 . . . . . . . . . 10  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
4021, 36, 393bitri 271 . . . . . . . . 9  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  1  =  0 )
417, 40sylib 196 . . . . . . . 8  |-  ( ( 1  +  0 )  =  0  ->  1  =  0 )
426, 41syl6 33 . . . . . . 7  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
1  =  0 ) )
4342necon3d 2606 . . . . . 6  |-  ( ( 1  +  c )  =  0  ->  (
1  =/=  0  -> 
c  =/=  0 ) )
443, 43mpi 17 . . . . 5  |-  ( ( 1  +  c )  =  0  ->  c  =/=  0 )
45 ax-rrecex 9475 . . . . 5  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
4644, 45sylan2 472 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
47 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  A  e.  CC )
48 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  RR )
4948recnd 9533 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  CC )
5047, 49mulcld 9527 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  x.  x )  e.  CC )
51 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  RR )
5251recnd 9533 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  CC )
5312a1i 11 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  CC )
5450, 52, 53adddid 9531 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( ( A  x.  x )  x.  c )  +  ( ( A  x.  x )  x.  0 ) ) )
5511a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  CC )
5655, 52, 53addassd 9529 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 1  +  ( c  +  0 ) ) )
57 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  c )  =  0 )
5857oveq1d 6211 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 0  +  0 ) )
5956, 58eqtr3d 2425 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 0  +  0 ) )
6034, 59, 573eqtr4a 2449 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 1  +  c ) )
6137a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  RR )
6251, 61readdcld 9534 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  e.  RR )
631a1i 11 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  RR )
64 readdcan 9665 . . . . . . . . . . 11  |-  ( ( ( c  +  0 )  e.  RR  /\  c  e.  RR  /\  1  e.  RR )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6562, 51, 63, 64syl3anc 1226 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6660, 65mpbid 210 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  =  c )
6766oveq2d 6212 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( A  x.  x )  x.  c ) )
6854, 67eqtr3d 2425 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( ( A  x.  x )  x.  c ) )
69 mul31 9659 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
7047, 49, 52, 69syl3anc 1226 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
71 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  x.  x )  =  1 )
7271oveq1d 6211 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( c  x.  x
)  x.  A )  =  ( 1  x.  A ) )
7347mulid2d 9525 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  x.  A )  =  A )
7470, 72, 733eqtrd 2427 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  A )
75 mul01 9670 . . . . . . . . 9  |-  ( ( A  x.  x )  e.  CC  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7650, 75syl 16 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7774, 76oveq12d 6214 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( A  + 
0 ) )
7868, 77, 743eqtr3d 2431 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  +  0 )  =  A )
7978exp42 609 . . . . 5  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( x  e.  RR  ->  ( (
c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) ) )
8079rexlimdv 2872 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( E. x  e.  RR  ( c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) )
8146, 80mpd 15 . . 3  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
8281rexlimiva 2870 . 2  |-  ( E. c  e.  RR  (
1  +  c )  =  0  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
831, 2, 82mp2b 10 1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   E.wrex 2733  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404   _ici 9405    + caddc 9406    x. cmul 9408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-ltxr 9544
This theorem is referenced by:  cnegex  9672  addid2  9674  addcan2  9676  addid1i  9678  addid1d  9691  subid  9751  subid1  9752  swrdccat3blem  12631  shftval3  12911  reim0  12953  isercolllem3  13491  fsumcvg  13536  summolem2a  13539  ovolicc1  22012  brbtwn2  24329  axsegconlem1  24341  ax5seglem4  24356  axeuclid  24387  axcontlem2  24389  axcontlem4  24391  risefac1  29321  stoweidlem26  31974  2zrngamnd  32947  aacllem  33550
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