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Mirrors > Home > MPE Home > Th. List > cnidOLD | Structured version Visualization version GIF version |
Description: Obsolete as of 23-Jan-2020. Use cnaddid 18480 instead. 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.) |
Ref | Expression |
---|---|
cnidOLD | ⊢ 0 = (GId‘ + ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnaddabloOLD 27776 | . . . 4 ⊢ + ∈ AbelOp | |
2 | ablogrpo 27741 | . . . 4 ⊢ ( + ∈ AbelOp → + ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ + ∈ GrpOp |
4 | ax-addf 10217 | . . . . . 6 ⊢ + :(ℂ × ℂ)⟶ℂ | |
5 | 4 | fdmi 6192 | . . . . 5 ⊢ dom + = (ℂ × ℂ) |
6 | 3, 5 | grporn 27715 | . . . 4 ⊢ ℂ = ran + |
7 | eqid 2771 | . . . 4 ⊢ (GId‘ + ) = (GId‘ + ) | |
8 | 6, 7 | grpoidval 27707 | . . 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 10421 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
11 | 10 | rgen 3071 | . . 3 ⊢ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 |
12 | 0cn 10234 | . . . 4 ⊢ 0 ∈ ℂ | |
13 | 6 | grpoideu 27703 | . . . . 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 6800 | . . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = (0 + 𝑥)) | |
16 | 15 | eqeq1d 2773 | . . . . . 6 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥)) |
17 | 16 | ralbidv 3135 | . . . . 5 ⊢ (𝑦 = 0 → (∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥)) |
18 | 17 | riota2 6776 | . . . 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 672 | . . 3 ⊢ (∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 ↔ (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0) |
20 | 11, 19 | mpbi 220 | . 2 ⊢ (℩𝑦 ∈ ℂ ∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥) = 0 |
21 | 9, 20 | eqtr2i 2794 | 1 ⊢ 0 = (GId‘ + ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃!wreu 3063 × cxp 5247 ‘cfv 6031 ℩crio 6753 (class class class)co 6793 ℂcc 10136 0cc0 10138 + caddc 10141 GrpOpcgr 27683 GIdcgi 27684 AbelOpcablo 27738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-addf 10217 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 df-sub 10470 df-neg 10471 df-grpo 27687 df-gid 27688 df-ablo 27739 |
This theorem is referenced by: cnnv 27872 |
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