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Mirrors > Home > MPE Home > Th. List > cnaddabl | Structured version Visualization version GIF version |
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 18738 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how Base and +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 20263. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnaddabl.g | ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} |
Ref | Expression |
---|---|
cnaddabl | ⊢ 𝐺 ∈ Abel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10410 | . . . 4 ⊢ ℂ ∈ V | |
2 | cnaddabl.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
3 | 2 | grpbase 16460 | . . . 4 ⊢ (ℂ ∈ V → ℂ = (Base‘𝐺)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℂ = (Base‘𝐺) |
5 | addex 12196 | . . . 4 ⊢ + ∈ V | |
6 | 2 | grpplusg 16461 | . . . 4 ⊢ ( + ∈ V → + = (+g‘𝐺)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐺) |
8 | addcl 10411 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
9 | addass 10416 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
10 | 0cn 10425 | . . 3 ⊢ 0 ∈ ℂ | |
11 | addid2 10617 | . . 3 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
12 | negcl 10680 | . . 3 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
13 | addcom 10620 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
14 | 12, 13 | mpdan 674 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
15 | negid 10728 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
16 | 14, 15 | eqtr3d 2810 | . . 3 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
17 | 4, 7, 8, 9, 10, 11, 12, 16 | isgrpi 17908 | . 2 ⊢ 𝐺 ∈ Grp |
18 | addcom 10620 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
19 | 17, 4, 7, 18 | isabli 18674 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 Vcvv 3409 {cpr 4437 〈cop 4441 ‘cfv 6182 (class class class)co 6970 ℂcc 10327 0cc0 10329 + caddc 10332 -cneg 10665 ndxcnx 16330 Basecbs 16333 +gcplusg 16415 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: cnaddinv 18741 cnaddcom 35553 |
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