Theorem addid0 8271

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 108	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> X e. CC )
2		simpr 109	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> Y e. CC )
3	1, 1, 2	subaddd 8227	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y <-> ( X + Y ) = X ) )
4		eqcom 2167	. . . . 5 |- ( ( X - X ) = Y <-> Y = ( X - X ) )
5		simpr 109	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = ( X - X ) )
6		subid 8117	. . . . . . . 8 |- ( X e. CC -> ( X - X ) = 0 )
7	6	adantr 274	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> ( X - X ) = 0 )
8	5, 7	eqtrd 2198	. . . . . 6 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = 0 )
9	8	ex 114	. . . . 5 |- ( X e. CC -> ( Y = ( X - X ) -> Y = 0 ) )
10	4, 9	syl5bi 151	. . . 4 |- ( X e. CC -> ( ( X - X ) = Y -> Y = 0 ) )
11	10	adantr 274	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y -> Y = 0 ) )
12	3, 11	sylbird 169	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X -> Y = 0 ) )
13		oveq2 5850	. . . . 5 |- ( Y = 0 -> ( X + Y ) = ( X + 0 ) )
14		addid1 8036	. . . . 5 |- ( X e. CC -> ( X + 0 ) = X )
15	13, 14	sylan9eqr 2221	. . . 4 |- ( ( X e. CC /\ Y = 0 ) -> ( X + Y ) = X )
16	15	ex 114	. . 3 |- ( X e. CC -> ( Y = 0 -> ( X + Y ) = X ) )
17	16	adantr 274	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( Y = 0 -> ( X + Y ) = X ) )
18	12, 17	impbid 128	1 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   0cc0 7753   + caddc 7756   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071

This theorem is referenced by:  addn0nid  8272