Theorem addid0 8292

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 8248	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y <-> ( X + Y ) = X ) )
4		eqcom 2172	. . . . 5 |- ( ( X - X ) = Y <-> Y = ( X - X ) )
5		simpr 109	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = ( X - X ) )
6		subid 8138	. . . . . . . 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 2203	. . . . . 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 5861	. . . . 5 |- ( Y = 0 -> ( X + Y ) = ( X + 0 ) )
14		addid1 8057	. . . . 5 |- ( X e. CC -> ( X + 0 ) = X )
15	13, 14	sylan9eqr 2225	. . . 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 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   0cc0 7774   + caddc 7777   - cmin 8090

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092

This theorem is referenced by:  addn0nid  8293