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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 18478 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 19983. (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 10219 | . . . 4 ⊢ ℂ ∈ V | |
2 | cnaddabl.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
3 | 2 | grpbase 16199 | . . . 4 ⊢ (ℂ ∈ V → ℂ = (Base‘𝐺)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℂ = (Base‘𝐺) |
5 | addex 12033 | . . . 4 ⊢ + ∈ V | |
6 | 2 | grpplusg 16200 | . . . 4 ⊢ ( + ∈ V → + = (+g‘𝐺)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐺) |
8 | addcl 10220 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
9 | addass 10225 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
10 | 0cn 10234 | . . 3 ⊢ 0 ∈ ℂ | |
11 | addid2 10421 | . . 3 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
12 | negcl 10483 | . . 3 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
13 | addcom 10424 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
14 | 12, 13 | mpdan 667 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
15 | negid 10530 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
16 | 14, 15 | eqtr3d 2807 | . . 3 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
17 | 4, 7, 8, 9, 10, 11, 12, 16 | isgrpi 17653 | . 2 ⊢ 𝐺 ∈ Grp |
18 | addcom 10424 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
19 | 17, 4, 7, 18 | isabli 18414 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 Vcvv 3351 {cpr 4318 〈cop 4322 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 0cc0 10138 + caddc 10141 -cneg 10469 ndxcnx 16061 Basecbs 16064 +gcplusg 16149 Abelcabl 18401 |
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-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-pre-mulgt0 10215 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-rmo 3069 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-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 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-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 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-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-plusg 16162 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-cmn 18402 df-abl 18403 |
This theorem is referenced by: cnaddinv 18481 cnaddcom 34781 |
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