Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnidOLD | Structured version Visualization version GIF version |
Description: Obsolete as of 23-Jan-2020. Use cnaddid 18273 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 27436 | . . . 4 ⊢ + ∈ AbelOp | |
2 | ablogrpo 27401 | . . . 4 ⊢ ( + ∈ AbelOp → + ∈ GrpOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ + ∈ GrpOp |
4 | ax-addf 10015 | . . . . . 6 ⊢ + :(ℂ × ℂ)⟶ℂ | |
5 | 4 | fdmi 6052 | . . . . 5 ⊢ dom + = (ℂ × ℂ) |
6 | 3, 5 | grporn 27375 | . . . 4 ⊢ ℂ = ran + |
7 | eqid 2622 | . . . 4 ⊢ (GId‘ + ) = (GId‘ + ) | |
8 | 6, 7 | grpoidval 27367 | . . 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 10219 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
11 | 10 | rgen 2922 | . . 3 ⊢ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥 |
12 | 0cn 10032 | . . . 4 ⊢ 0 ∈ ℂ | |
13 | 6 | grpoideu 27363 | . . . . 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 6657 | . . . . . . 7 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = (0 + 𝑥)) | |
16 | 15 | eqeq1d 2624 | . . . . . 6 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥)) |
17 | 16 | ralbidv 2986 | . . . . 5 ⊢ (𝑦 = 0 → (∀𝑥 ∈ ℂ (𝑦 + 𝑥) = 𝑥 ↔ ∀𝑥 ∈ ℂ (0 + 𝑥) = 𝑥)) |
18 | 17 | riota2 6633 | . . . 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 708 | . . 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 2645 | 1 ⊢ 0 = (GId‘ + ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 × cxp 5112 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 GrpOpcgr 27343 GIdcgi 27344 AbelOpcablo 27398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ablo 27399 |
This theorem is referenced by: cnnv 27532 |
Copyright terms: Public domain | W3C validator |