Theorem addlid 11420

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 11418	. 2 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
2		cnegex 11418	. . . 4 ⊢ (𝑥 ∈ ℂ → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
3	2	ad2antrl 741	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0)
4		0cn 11225	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
5		addass 11214	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
6	4, 4, 5	mp3an12 1480	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
7	6	adantr 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
8	7	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = (0 + (0 + 𝑦)))
9		00id 11412	. . . . . . . . 9 ⊢ (0 + 0) = 0
10	9	oveq1i 7426	. . . . . . . 8 ⊢ ((0 + 0) + 𝑦) = (0 + 𝑦)
11		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝐴 ∈ ℂ)
12		simp2l 1218	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑥 ∈ ℂ)
13		simp3l 1220	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → 𝑦 ∈ ℂ)
14	11, 12, 13	addassd 11258	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (𝐴 + (𝑥 + 𝑦)))
15		simp2r 1219	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 𝑥) = 0)
16	15	oveq1d 7431	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((𝐴 + 𝑥) + 𝑦) = (0 + 𝑦))
17		simp3r 1221	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝑥 + 𝑦) = 0)
18	17	oveq2d 7432	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + (𝑥 + 𝑦)) = (𝐴 + 0))
19	14, 16, 18	3eqtr3rd 2806	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = (0 + 𝑦))
20		addrid 11417	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
21	20	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (𝐴 + 0) = 𝐴)
22	19, 21	eqtr3d 2799	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝑦) = 𝐴)
23	10, 22	eqtrid 2809	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → ((0 + 0) + 𝑦) = 𝐴)
24	22	oveq2d 7432	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + (0 + 𝑦)) = (0 + 𝐴))
25	8, 23, 24	3eqtr3rd 2806	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0) ∧ (𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0)) → (0 + 𝐴) = 𝐴)
26	25	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → ((𝑦 ∈ ℂ ∧ (𝑥 + 𝑦) = 0) → (0 + 𝐴) = 𝐴))
27	26	expd 421	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (𝑦 ∈ ℂ → ((𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴)))
28	27	rexlimdv 3163	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (∃𝑦 ∈ ℂ (𝑥 + 𝑦) = 0 → (0 + 𝐴) = 𝐴))
29	3, 28	mpd 16	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ (𝐴 + 𝑥) = 0)) → (0 + 𝐴) = 𝐴)
30	1, 29	rexlimddv 3171	1 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  (class class class)co 7416  ℂcc 11125  0cc0 11127   + caddc 11130

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-ltxr 11275

This theorem is used by:  addcan  11421  addlidi  11425  addlidd  11438  negneg  11535  fzo0addel  13776  fzoaddel2  13778  divfl0  13887  modid  13959  modsumfzodifsn  14010  swrdspsleq  14737  swrds1  14738  isercolllem3  15756  sumrblem  15799  summolem2a  15803  fsum0diag2  15871  eftlub  16201  gcdid  16621  chnccat  18718  cnaddablx  19996  cnaddabl  19997  cnaddid  19998  cncrng  21607  cnlmod  25369  ptolemy  26731  logtayl  26895  leibpilem2  27176  axcontlem2  29408  cnaddabloOLD  31048  cnidOLD  31049  gsumzrsum  33492  dvcosax  46741  2zrngamnd  49149  aacllem  50759