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Mirrors > Home > MPE Home > Th. List > addcom | Structured version Visualization version GIF version |
Description: Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
addcom | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 10371 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 1 ∈ ℂ) | |
2 | 1, 1 | addcld 10396 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 + 1) ∈ ℂ) |
3 | simpl 476 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | simpr 479 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 2, 3, 4 | adddid 10401 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 1) · (𝐴 + 𝐵)) = (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵))) |
6 | 3, 4 | addcld 10396 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
7 | 1p1times 10547 | . . . . . . 7 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((1 + 1) · (𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 1) · (𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
9 | 1p1times 10547 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
10 | 1p1times 10547 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((1 + 1) · 𝐵) = (𝐵 + 𝐵)) | |
11 | 9, 10 | oveqan12d 6941 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵)) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
12 | 5, 8, 11 | 3eqtr3rd 2823 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + (𝐵 + 𝐵)) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
13 | 3, 3 | addcld 10396 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐴) ∈ ℂ) |
14 | 13, 4, 4 | addassd 10399 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
15 | 6, 3, 4 | addassd 10399 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
16 | 12, 14, 15 | 3eqtr4d 2824 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵)) |
17 | 13, 4 | addcld 10396 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) ∈ ℂ) |
18 | 6, 3 | addcld 10396 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) ∈ ℂ) |
19 | addcan2 10561 | . . . . 5 ⊢ ((((𝐴 + 𝐴) + 𝐵) ∈ ℂ ∧ ((𝐴 + 𝐵) + 𝐴) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵) ↔ ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴))) | |
20 | 17, 18, 4, 19 | syl3anc 1439 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵) ↔ ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴))) |
21 | 16, 20 | mpbid 224 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴)) |
22 | 3, 3, 4 | addassd 10399 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = (𝐴 + (𝐴 + 𝐵))) |
23 | 3, 4, 3 | addassd 10399 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) = (𝐴 + (𝐵 + 𝐴))) |
24 | 21, 22, 23 | 3eqtr3d 2822 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴))) |
25 | 4, 3 | addcld 10396 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) ∈ ℂ) |
26 | addcan 10560 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℂ ∧ (𝐵 + 𝐴) ∈ ℂ) → ((𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴)) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) | |
27 | 3, 6, 25, 26 | syl3anc 1439 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴)) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
28 | 24, 27 | mpbid 224 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 1c1 10273 + caddc 10275 · cmul 10277 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 |
This theorem is referenced by: addcomi 10567 ltaddnegr 10592 add12 10593 add32 10594 add42 10597 subsub23 10627 pncan2 10629 addsub 10634 addsub12 10636 addsubeq4 10638 sub32 10657 pnpcan2 10663 ppncan 10665 sub4 10668 negsubdi2 10682 ltaddsub2 10850 leaddsub2 10852 leltadd 10859 ltaddpos2 10866 addge02 10886 conjmul 11092 recp1lt1 11275 recreclt 11276 avgle1 11622 avgle2 11623 avgle 11624 nn0nnaddcl 11675 xaddcom 12383 fzen 12675 fzshftral 12746 fzo0addelr 12842 elfzoext 12844 flzadd 12946 addmodidr 13038 modadd2mod 13039 nn0ennn 13097 seradd 13161 bernneq2 13310 ccatalpha 13683 revccat 13912 2cshwcom 13967 shftval2 14222 shftval4 14224 crim 14262 absmax 14476 climshft2 14721 summolem3 14852 binom1dif 14969 isumshft 14975 arisum 14996 mertenslem1 15019 bpolydiflem 15187 addcos 15306 demoivreALT 15333 dvdsaddr 15432 sumodd 15518 divalglem4 15526 divalgb 15534 gcdaddm 15652 hashdvds 15884 phiprmpw 15885 pythagtriplem2 15926 prmgaplem7 16165 mulgnndir 17955 cnaddablx 18657 cnaddabl 18658 zaddablx 18661 cncrng 20163 ioo2bl 23004 icopnfcnv 23149 uniioombllem3 23789 fta1glem1 24362 plyremlem 24496 fta1lem 24499 vieta1lem1 24502 vieta1lem2 24503 aaliou3lem2 24535 dvradcnv 24612 pserdv2 24621 reeff1olem 24637 ptolemy 24686 logcnlem4 24828 cxpsqrt 24886 atandm2 25055 atandm4 25057 atanlogsublem 25093 2efiatan 25096 dvatan 25113 birthdaylem2 25131 emcllem2 25175 fsumharmonic 25190 wilthlem1 25246 wilthlem2 25247 basellem8 25266 1sgmprm 25376 perfectlem2 25407 pntibndlem1 25730 pntibndlem2 25732 pntlemd 25735 pntlemc 25736 eucrctshift 27647 cnaddabloOLD 28008 cdj3lem3b 29871 isarchi3 30303 archiabllem2c 30311 cos2h 34027 tan2h 34028 eldioph2lem1 38287 addcomgi 39618 fz0addcom 42363 epoo 42641 perfectALTVlem2 42660 sbgoldbaltlem2 42697 |
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