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Theorem cnidOLD 28368
Description: Obsolete version of cnaddid 18986. 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.
StepHypRef Expression
1 cnaddabloOLD 28367 . . . 4 + ∈ AbelOp
2 ablogrpo 28333 . . . 4 ( + ∈ AbelOp → + ∈ GrpOp)
31, 2ax-mp 5 . . 3 + ∈ GrpOp
4 ax-addf 10609 . . . . . 6 + :(ℂ × ℂ)⟶ℂ
54fdmi 6502 . . . . 5 dom + = (ℂ × ℂ)
63, 5grporn 28307 . . . 4 ℂ = ran +
7 eqid 2801 . . . 4 (GId‘ + ) = (GId‘ + )
86, 7grpoidval 28299 . . 3 ( + ∈ GrpOp → (GId‘ + ) = (𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥))
93, 8ax-mp 5 . 2 (GId‘ + ) = (𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥)
10 addid2 10816 . . . 4 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
1110rgen 3119 . . 3 𝑥 ∈ ℂ (0 + 𝑥) = 𝑥
12 0cn 10626 . . . 4 0 ∈ ℂ
136grpoideu 28295 . . . . 5 ( + ∈ GrpOp → ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥)
143, 13ax-mp 5 . . . 4 ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥
15 oveq1 7146 . . . . . . 7 (𝑦 = 0 → (𝑦 + 𝑥) = (0 + 𝑥))
1615eqeq1d 2803 . . . . . 6 (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥))
1716ralbidv 3165 . . . . 5 (𝑦 = 0 → (∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥))
1817riota2 7122 . . . 4 ((0 ∈ ℂ ∧ ∃!𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) → (∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 ↔ (𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0))
1912, 14, 18mp2an 691 . . 3 (∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 ↔ (𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0)
2011, 19mpbi 233 . 2 (𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0
219, 20eqtr2i 2825 1 0 = (GId‘ + )
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  wral 3109  ∃!wreu 3111   × cxp 5521  cfv 6328  crio 7096  (class class class)co 7139  cc 10528  0cc0 10530   + caddc 10533  GrpOpcgr 28275  GIdcgi 28276  AbelOpcablo 28330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-addf 10609
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-ltxr 10673  df-sub 10865  df-neg 10866  df-grpo 28279  df-gid 28280  df-ablo 28331
This theorem is referenced by:  cnnv  28463
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