Theorem addid0 8392

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 8348	. . 3 |- ( ( X e. CC /\ Y e. CC ) -> ( ( X - X ) = Y <-> ( X + Y ) = X ) )
4		eqcom 2195	. . . . 5 |- ( ( X - X ) = Y <-> Y = ( X - X ) )
5		simpr 110	. . . . . . 7 |- ( ( X e. CC /\ Y = ( X - X ) ) -> Y = ( X - X ) )
6		subid 8238	. . . . . . . 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 2226	. . . . . 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 5926	. . . . 5 |- ( Y = 0 -> ( X + Y ) = ( X + 0 ) )
14		addrid 8157	. . . . 5 |- ( X e. CC -> ( X + 0 ) = X )
15	13, 14	sylan9eqr 2248	. . . 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 1364   e. wcel 2164  (class class class)co 5918   CCcc 7870   0cc0 7872   + caddc 7875   - cmin 8190

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192

This theorem is referenced by:  addn0nid  8393