Theorem addid0 7477

Description: If adding a number to a another number yields the other number, the added number must be 0. This shows that 0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)

Assertion

Ref	Expression
addid0	|- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) )

Proof of Theorem addid0

Step	Hyp	Ref	Expression
1		simpl 107	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> X e. CC )
2		simpr 108	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> Y e. CC )
3	1, 1, 2	subaddd 7437	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y <-> ( X + Y ) = X ) )
4		eqcom 2083	. . . . 5 |- ( ( X - X ) = Y <-> Y = ( X - X ) )
5		simpr 108	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = ( X - X ) )
6		subid 7327	. . . . . . . 8 |- ( X e. CC -> ( X - X ) = 0 )
7	6	adantr 270	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> ( X - X ) = 0 )
8	5, 7	eqtrd 2113	. . . . . 6 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = 0 )
9	8	ex 113	. . . . 5 |- ( X e. CC -> ( Y = ( X - X ) -> Y = 0 ) )
10	4, 9	syl5bi 150	. . . 4 |- ( X e. CC -> ( ( X - X ) = Y -> Y = 0 ) )
11	10	adantr 270	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y -> Y = 0 ) )
12	3, 11	sylbird 168	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X -> Y = 0 ) )
13		oveq2 5540	. . . . 5 |- ( Y = 0 -> ( X + Y ) = ( X + 0 ) )
14		addid1 7246	. . . . 5 |- ( X e. CC -> ( X + 0 ) = X )
15	13, 14	sylan9eqr 2135	. . . 4 |- ( ( X e. CC /\ Y = 0 ) -> ( X + Y ) = X )
16	15	ex 113	. . 3 |- ( X e. CC -> ( Y = 0 -> ( X + Y ) = X ) )
17	16	adantr 270	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( Y = 0 -> ( X + Y ) = X ) )
18	12, 17	impbid 127	1 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   0cc0 6981   + caddc 6984   - cmin 7279

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sub 7281

This theorem is referenced by:  addn0nid  7478