Theorem addid0 8332

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 109	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> X e. CC )
2		simpr 110	. . . 4 |- ( ( X e. CC /\ Y e. CC ) -> Y e. CC )
3	1, 1, 2	subaddd 8288	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y <-> ( X + Y ) = X ) )
4		eqcom 2179	. . . . 5 |- ( ( X - X ) = Y <-> Y = ( X - X ) )
5		simpr 110	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = ( X - X ) )
6		subid 8178	. . . . . . . 8 |- ( X e. CC -> ( X - X ) = 0 )
7	6	adantr 276	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> ( X - X ) = 0 )
8	5, 7	eqtrd 2210	. . . . . 6 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = 0 )
9	8	ex 115	. . . . 5 |- ( X e. CC -> ( Y = ( X - X ) -> Y = 0 ) )
10	4, 9	biimtrid 152	. . . 4 |- ( X e. CC -> ( ( X - X ) = Y -> Y = 0 ) )
11	10	adantr 276	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y -> Y = 0 ) )
12	3, 11	sylbird 170	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X -> Y = 0 ) )
13		oveq2 5885	. . . . 5 |- ( Y = 0 -> ( X + Y ) = ( X + 0 ) )
14		addid1 8097	. . . . 5 |- ( X e. CC -> ( X + 0 ) = X )
15	13, 14	sylan9eqr 2232	. . . 4 |- ( ( X e. CC /\ Y = 0 ) -> ( X + Y ) = X )
16	15	ex 115	. . 3 |- ( X e. CC -> ( Y = 0 -> ( X + Y ) = X ) )
17	16	adantr 276	. 2 |- ( ( X e. CC /\ Y e. CC ) -> ( Y = 0 -> ( X + Y ) = X ) )
18	12, 17	impbid 129	1 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X + Y ) = X <-> Y = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   0cc0 7813   + caddc 7816   - cmin 8130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132

This theorem is referenced by:  addn0nid  8333