Theorem addlid 11317

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
addlid	⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Proof of Theorem addlid

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

Step	Hyp	Ref	Expression
1		cnegex 11315	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
2		cnegex 11315	. . . 4 ⊢ (𝑥 ∈ ℂ → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
3	2	ad2antrl 728	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
4		0cn 11126	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
5		addass 11115	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
6	4, 4, 5	mp3an12 1453	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
8	7	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
9		00id 11309	. . . . . . . . 9 ⊢ (0 + 0) = 0
10	9	oveq1i 7363	. . . . . . . 8 ⊢ ((0 + 0) + 𝑦) = (0 + 𝑦)
11		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝐴 ∈ ℂ)
12		simp2l 1200	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑥 ∈ ℂ)
13		simp3l 1202	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑦 ∈ ℂ)
14	11, 12, 13	addassd 11156	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (𝐴 + (𝑥 + 𝑦)))
15		simp2r 1201	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 𝑥) = 0)
16	15	oveq1d 7368	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (0 + 𝑦))
17		simp3r 1203	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝑥 + 𝑦) = 0)
18	17	oveq2d 7369	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + (𝑥 + 𝑦)) = (𝐴 + 0))
19	14, 16, 18	3eqtr3rd 2773	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = (0 + 𝑦))
20		addrid 11314	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
21	20	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = 𝐴)
22	19, 21	eqtr3d 2766	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝑦) = 𝐴)
23	10, 22	eqtrid 2776	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = 𝐴)
24	22	oveq2d 7369	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + (0 + 𝑦)) = (0 + 𝐴))
25	8, 23, 24	3eqtr3rd 2773	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝐴) = 𝐴)
26	25	3expia 1121	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → (0 + 𝐴) = 𝐴))
27	26	expd 415	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (𝑦 ∈ ℂ → ((𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴)))
28	27	rexlimdv 3128	. . 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 3136	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  (class class class)co 7353  ℂcc 11026  0cc0 11028   + caddc 11031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-po 5531  df-so 5532  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-ltxr 11173

This theorem is referenced by:  addcan  11318  addlidi  11322  addlidd  11335  negneg  11432  fzo0addel  13639  fzoaddel2  13641  divfl0  13746  modid  13818  modsumfzodifsn  13869  swrdspsleq  14590  swrds1  14591  isercolllem3  15592  sumrblem  15636  summolem2a  15640  fsum0diag2  15708  eftlub  16036  gcdid  16456  cnaddablx  19765  cnaddabl  19766  cnaddid  19767  cncrng  21313  cncrngOLD  21314  cnlmod  25056  ptolemy  26421  logtayl  26585  leibpilem2  26867  axcontlem2  28928  cnaddabloOLD  30543  cnidOLD  30544  gsumzrsum  33025  dvcosax  45908  2zrngamnd  48219  aacllem  49774