![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnaddablx | Structured version Visualization version GIF version |
Description: The complex numbers are an Abelian group under addition. This version of cnaddabl 18739 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 18739 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.) |
Ref | Expression |
---|---|
cnaddablx.g | ⊢ 𝐺 = {〈1, ℂ〉, 〈2, + 〉} |
Ref | Expression |
---|---|
cnaddablx | ⊢ 𝐺 ∈ Abel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10410 | . . 3 ⊢ ℂ ∈ V | |
2 | addex 12196 | . . 3 ⊢ + ∈ V | |
3 | cnaddablx.g | . . 3 ⊢ 𝐺 = {〈1, ℂ〉, 〈2, + 〉} | |
4 | addcl 10411 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
5 | addass 10416 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
6 | 0cn 10425 | . . 3 ⊢ 0 ∈ ℂ | |
7 | addid2 10617 | . . 3 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
8 | negcl 10680 | . . 3 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
9 | addcom 10620 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
10 | 8, 9 | mpdan 674 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
11 | negid 10728 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
12 | 10, 11 | eqtr3d 2810 | . . 3 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 12 | isgrpix 17912 | . 2 ⊢ 𝐺 ∈ Grp |
14 | 1, 2, 3 | grpbasex 16463 | . 2 ⊢ ℂ = (Base‘𝐺) |
15 | 1, 2, 3 | grpplusgx 16464 | . 2 ⊢ + = (+g‘𝐺) |
16 | addcom 10620 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
17 | 13, 14, 15, 16 | isabli 18674 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 {cpr 4437 〈cop 4441 (class class class)co 6970 ℂcc 10327 0cc0 10329 1c1 10330 + caddc 10332 -cneg 10665 2c2 11489 Abelcabl 18661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-addf 10408 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-nn 11434 df-2 11497 df-n0 11702 df-z 11788 df-uz 12053 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-plusg 16428 df-0g 16565 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-grp 17888 df-cmn 18662 df-abl 18663 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |