Theorem addlid 11473

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 11471	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
2		cnegex 11471	. . . 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 11282	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
5		addass 11271	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
6	4, 4, 5	mp3an12 1451	. . . . . . . . 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 11465	. . . . . . . . 9 ⊢ (0 + 0) = 0
10	9	oveq1i 7458	. . . . . . . 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 1199	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑥 ∈ ℂ)
13		simp3l 1201	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑦 ∈ ℂ)
14	11, 12, 13	addassd 11312	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (𝐴 + (𝑥 + 𝑦)))
15		simp2r 1200	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 𝑥) = 0)
16	15	oveq1d 7463	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (0 + 𝑦))
17		simp3r 1202	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝑥 + 𝑦) = 0)
18	17	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + (𝑥 + 𝑦)) = (𝐴 + 0))
19	14, 16, 18	3eqtr3rd 2789	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = (0 + 𝑦))
20		addrid 11470	. . . . . . . . . 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 2782	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝑦) = 𝐴)
23	10, 22	eqtrid 2792	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = 𝐴)
24	22	oveq2d 7464	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + (0 + 𝑦)) = (0 + 𝐴))
25	8, 23, 24	3eqtr3rd 2789	. . . . . 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 3159	. . 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 3167	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  (class class class)co 7448  ℂcc 11182  0cc0 11184   + caddc 11187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329

This theorem is referenced by:  addcan  11474  addlidi  11478  addlidd  11491  negneg  11586  fzo0addel  13770  fzoaddel2  13772  divfl0  13875  modid  13947  modsumfzodifsn  13995  swrdspsleq  14713  swrds1  14714  isercolllem3  15715  sumrblem  15759  summolem2a  15763  fsum0diag2  15831  eftlub  16157  gcdid  16573  cnaddablx  19910  cnaddabl  19911  cnaddid  19912  cncrng  21424  cncrngOLD  21425  cnlmod  25192  ptolemy  26556  logtayl  26720  leibpilem2  27002  axcontlem2  28998  cnaddabloOLD  30613  cnidOLD  30614  dvcosax  45847  2zrngamnd  47970  aacllem  48895