Theorem cnidOLD 28359

Description: Obsolete version of cnaddid 18990. The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
cnidOLD	⊢ 0 = (GId‘ + )

Proof of Theorem cnidOLD

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

Step	Hyp	Ref	Expression
1		cnaddabloOLD 28358	. . . 4 ⊢ + ∈ AbelOp
2		ablogrpo 28324	. . . 4 ⊢ ( + ∈ AbelOp → + ∈ GrpOp)
3	1, 2	ax-mp 5	. . 3 ⊢ + ∈ GrpOp
4		ax-addf 10616	. . . . . 6 ⊢ + :(ℂ × ℂ)⟶ℂ
5	4	fdmi 6524	. . . . 5 ⊢ dom + = (ℂ × ℂ)
6	3, 5	grporn 28298	. . . 4 ⊢ ℂ = ran +
7		eqid 2821	. . . 4 ⊢ (GId‘ + ) = (GId‘ + )
8	6, 7	grpoidval 28290	. . 3 ⊢ ( + ∈ GrpOp → (GId‘ + ) = (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥))
9	3, 8	ax-mp 5	. 2 ⊢ (GId‘ + ) = (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥)
10		addid2 10823	. . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
11	10	rgen 3148	. . 3 ⊢ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥
12		0cn 10633	. . . 4 ⊢ 0 ∈ ℂ
13	6	grpoideu 28286	. . . . 5 ⊢ ( + ∈ GrpOp → ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥)
14	3, 13	ax-mp 5	. . . 4 ⊢ ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥
15		oveq1 7163	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = (0 + 𝑥))
16	15	eqeq1d 2823	. . . . . 6 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥))
17	16	ralbidv 3197	. . . . 5 ⊢ (𝑦 = 0 → (∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥))
18	17	riota2 7139	. . . 4 ⊢ ((0 ∈ ℂ ∧ ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) → (∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0))
19	12, 14, 18	mp2an 690	. . 3 ⊢ (∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0)
20	11, 19	mpbi 232	. 2 ⊢ (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0
21	9, 20	eqtr2i 2845	1 ⊢ 0 = (GId‘ + )

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃!wreu 3140   × cxp 5553  ‘cfv 6355  ℩crio 7113  (class class class)co 7156  ℂcc 10535  0cc0 10537   + caddc 10540  GrpOpcgr 28266  GIdcgi 28267  AbelOpcablo 28321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-addf 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-ltxr 10680  df-sub 10872  df-neg 10873  df-grpo 28270  df-gid 28271  df-ablo 28322

This theorem is referenced by:  cnnv  28454