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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnaddcom | Structured version Visualization version GIF version |
Description: Recover the commutative law of addition for complex numbers from the Abelian group structure. (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnaddcom | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
2 | 1 | cnaddabl 19470 | . 2 ⊢ {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∈ Abel |
3 | cnex 10952 | . . . 4 ⊢ ℂ ∈ V | |
4 | 1 | grpbase 16996 | . . . 4 ⊢ (ℂ ∈ V → ℂ = (Base‘{〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉})) |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ℂ = (Base‘{〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉}) |
6 | addex 12728 | . . . 4 ⊢ + ∈ V | |
7 | 1 | grpplusg 16998 | . . . 4 ⊢ ( + ∈ V → + = (+g‘{〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉})) |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ + = (+g‘{〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉}) |
9 | 5, 8 | ablcom 19404 | . 2 ⊢ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∈ Abel ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
10 | 2, 9 | mp3an1 1447 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 {cpr 4563 〈cop 4567 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 + caddc 10874 ndxcnx 16894 Basecbs 16912 +gcplusg 16962 Abelcabl 19387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-cmn 19388 df-abl 19389 |
This theorem is referenced by: (None) |
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