Theorem addid2 10098

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 10096	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
2		cnegex 10096	. . . 4 ⊢ (𝑥 ∈ ℂ → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
3	2	ad2antrl 760	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
4		0cn 9911	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
5		addass 9902	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
6	4, 4, 5	mp3an12 1406	. . . . . . . . 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 1077	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
9		00id 10090	. . . . . . . . 9 ⊢ (0 + 0) = 0
10	9	oveq1i 6559	. . . . . . . 8 ⊢ ((0 + 0) + 𝑦) = (0 + 𝑦)
11		simp1 1054	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝐴 ∈ ℂ)
12		simp2l 1080	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑥 ∈ ℂ)
13		simp3l 1082	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑦 ∈ ℂ)
14	11, 12, 13	addassd 9941	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (𝐴 + (𝑥 + 𝑦)))
15		simp2r 1081	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 𝑥) = 0)
16	15	oveq1d 6564	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (0 + 𝑦))
17		simp3r 1083	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝑥 + 𝑦) = 0)
18	17	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + (𝑥 + 𝑦)) = (𝐴 + 0))
19	14, 16, 18	3eqtr3rd 2653	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = (0 + 𝑦))
20		addid1 10095	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
21	20	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = 𝐴)
22	19, 21	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝑦) = 𝐴)
23	10, 22	syl5eq 2656	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = 𝐴)
24	22	oveq2d 6565	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + (0 + 𝑦)) = (0 + 𝐴))
25	8, 23, 24	3eqtr3rd 2653	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝐴) = 𝐴)
26	25	3expia 1259	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → (0 + 𝐴) = 𝐴))
27	26	expd 451	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (𝑦 ∈ ℂ → ((𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴)))
28	27	rexlimdv 3012	. . 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 3017	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958

This theorem is referenced by:  addcan  10099  addid2i  10103  addid2d  10116  negneg  10210  fz0to4untppr  12311  fzo0addel  12389  fzoaddel2  12391  divfl0  12487  modid  12557  modsumfzodifsn  12605  swrdspsleq  13301  swrds1  13303  isercolllem3  14245  sumrblem  14289  summolem2a  14293  fsum0diag2  14357  eftlub  14678  gcdid  15086  cnaddablx  18094  cnaddabl  18095  cnaddid  18096  cncrng  19586  cnlmod  22748  ptolemy  24052  logtayl  24206  leibpilem2  24468  axcontlem2  25645  cnaddabloOLD  26820  cnidOLD  26821  dvcosax  38816  2zrngamnd  41731  aacllem  42356