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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 11241 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 1 ∈ ℂ) | |
2 | 1, 1 | addcld 11265 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 + 1) ∈ ℂ) |
3 | simpl 481 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | simpr 483 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 2, 3, 4 | adddid 11270 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 1) · (𝐴 + 𝐵)) = (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵))) |
6 | 3, 4 | addcld 11265 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
7 | 1p1times 11417 | . . . . . . 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 11417 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
10 | 1p1times 11417 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((1 + 1) · 𝐵) = (𝐵 + 𝐵)) | |
11 | 9, 10 | oveqan12d 7438 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵)) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
12 | 5, 8, 11 | 3eqtr3rd 2774 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + (𝐵 + 𝐵)) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
13 | 3, 3 | addcld 11265 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐴) ∈ ℂ) |
14 | 13, 4, 4 | addassd 11268 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
15 | 6, 3, 4 | addassd 11268 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
16 | 12, 14, 15 | 3eqtr4d 2775 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵)) |
17 | 13, 4 | addcld 11265 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) ∈ ℂ) |
18 | 6, 3 | addcld 11265 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) ∈ ℂ) |
19 | addcan2 11431 | . . . . 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 1368 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵) ↔ ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴))) |
21 | 16, 20 | mpbid 231 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴)) |
22 | 3, 3, 4 | addassd 11268 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = (𝐴 + (𝐴 + 𝐵))) |
23 | 3, 4, 3 | addassd 11268 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) = (𝐴 + (𝐵 + 𝐴))) |
24 | 21, 22, 23 | 3eqtr3d 2773 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴))) |
25 | 4, 3 | addcld 11265 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) ∈ ℂ) |
26 | addcan 11430 | . . 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 1368 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴)) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
28 | 24, 27 | mpbid 231 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 |
This theorem is referenced by: addcomi 11437 ltaddnegr 11462 add12 11463 add32 11464 add42 11467 subsub23 11497 pncan2 11499 addsub 11503 addsub12 11505 addsubeq4 11507 sub32 11526 pnpcan2 11532 ppncan 11534 sub4 11537 negsubdi2 11551 ltaddsub2 11721 leaddsub2 11723 leltadd 11730 ltaddpos2 11737 addge02 11757 conjmul 11964 recp1lt1 12145 recreclt 12146 avgle1 12485 avgle2 12486 avgle 12487 nn0nnaddcl 12536 xaddcom 13254 fzen 13553 fzshftral 13624 fzo0addelr 13722 elfzoext 13724 flzadd 13827 addmodidr 13921 modadd2mod 13922 nn0ennn 13980 seradd 14045 bernneq2 14228 ccatrn 14575 ccatalpha 14579 revccat 14752 2cshwcom 14802 shftval2 15058 shftval4 15060 crim 15098 absmax 15312 climshft2 15562 summolem3 15696 binom1dif 15815 isumshft 15821 arisum 15842 mertenslem1 15866 bpolydiflem 16034 addcos 16154 demoivreALT 16181 dvdsaddr 16283 sumodd 16368 divalglem4 16376 divalgb 16384 gcdaddm 16503 hashdvds 16747 phiprmpw 16748 pythagtriplem2 16789 prmgaplem7 17029 mulgnndir 19066 cnaddablx 19835 cnaddabl 19836 zaddablx 19839 cncrngOLD 21334 ioo2bl 24753 icopnfcnv 24911 uniioombllem3 25558 fta1glem1 26147 plyremlem 26284 fta1lem 26287 vieta1lem1 26290 vieta1lem2 26291 aaliou3lem2 26323 dvradcnv 26402 pserdv2 26412 reeff1olem 26428 ptolemy 26476 logcnlem4 26624 cxpsqrt 26682 atandm2 26854 atandm4 26856 atanlogsublem 26892 2efiatan 26895 dvatan 26912 birthdaylem2 26929 emcllem2 26974 fsumharmonic 26989 wilthlem1 27045 wilthlem2 27046 basellem8 27065 1sgmprm 27177 perfectlem2 27208 pntibndlem1 27567 pntibndlem2 27569 pntlemd 27572 pntlemc 27573 eucrctshift 30125 cnaddabloOLD 30463 cdj3lem3b 32322 isarchi3 32987 archiabllem2c 32995 cos2h 37215 tan2h 37216 lcmineqlem4 41635 2xp3dxp2ge1d 41827 eldioph2lem1 42322 addcomgi 44035 fz0addcom 46835 epoo 47180 perfectALTVlem2 47199 sbgoldbaltlem2 47257 |
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