Theorem addid2 10812

Description: 0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
addid2	⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Proof of Theorem addid2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnegex 10810	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
2		cnegex 10810	. . . 4 ⊢ (𝑥 ∈ ℂ → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
3	2	ad2antrl 727	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
4		0cn 10622	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
5		addass 10613	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
6	4, 4, 5	mp3an12 1448	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
7	6	adantr 484	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
8	7	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
9		00id 10804	. . . . . . . . 9 ⊢ (0 + 0) = 0
10	9	oveq1i 7145	. . . . . . . 8 ⊢ ((0 + 0) + 𝑦) = (0 + 𝑦)
11		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝐴 ∈ ℂ)
12		simp2l 1196	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑥 ∈ ℂ)
13		simp3l 1198	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑦 ∈ ℂ)
14	11, 12, 13	addassd 10652	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (𝐴 + (𝑥 + 𝑦)))
15		simp2r 1197	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 𝑥) = 0)
16	15	oveq1d 7150	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (0 + 𝑦))
17		simp3r 1199	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝑥 + 𝑦) = 0)
18	17	oveq2d 7151	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + (𝑥 + 𝑦)) = (𝐴 + 0))
19	14, 16, 18	3eqtr3rd 2842	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = (0 + 𝑦))
20		addid1 10809	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
21	20	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = 𝐴)
22	19, 21	eqtr3d 2835	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝑦) = 𝐴)
23	10, 22	syl5eq 2845	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = 𝐴)
24	22	oveq2d 7151	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + (0 + 𝑦)) = (0 + 𝐴))
25	8, 23, 24	3eqtr3rd 2842	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝐴) = 𝐴)
26	25	3expia 1118	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → (0 + 𝐴) = 𝐴))
27	26	expd 419	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (𝑦 ∈ ℂ → ((𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴)))
28	27	rexlimdv 3242	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴))
29	3, 28	mpd 15	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (0 + 𝐴) = 𝐴)
30	1, 29	rexlimddv 3250	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  (class class class)co 7135  ℂcc 10524  0cc0 10526   + caddc 10529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-ltxr 10669

This theorem is referenced by:  addcan  10813  addid2i  10817  addid2d  10830  negneg  10925  fz0to4untppr  13005  fzo0addel  13086  fzoaddel2  13088  divfl0  13189  modid  13259  modsumfzodifsn  13307  swrdspsleq  14018  swrds1  14019  isercolllem3  15015  sumrblem  15060  summolem2a  15064  fsum0diag2  15130  eftlub  15454  gcdid  15865  cnaddablx  18981  cnaddabl  18982  cnaddid  18983  cncrng  20112  cnlmod  23745  ptolemy  25089  logtayl  25251  leibpilem2  25527  axcontlem2  26759  cnaddabloOLD  28364  cnidOLD  28365  dvcosax  42568  2zrngamnd  44565  aacllem  45329