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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 9935 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 1 ∈ ℂ) | |
2 | 1, 1 | addcld 9938 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (1 + 1) ∈ ℂ) |
3 | simpl 472 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | simpr 476 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
5 | 2, 3, 4 | adddid 9943 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 1) · (𝐴 + 𝐵)) = (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵))) |
6 | 3, 4 | addcld 9938 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
7 | 1p1times 10086 | . . . . . . 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 10086 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | |
10 | 1p1times 10086 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((1 + 1) · 𝐵) = (𝐵 + 𝐵)) | |
11 | 9, 10 | oveqan12d 6568 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((1 + 1) · 𝐴) + ((1 + 1) · 𝐵)) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
12 | 5, 8, 11 | 3eqtr3rd 2653 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + (𝐵 + 𝐵)) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
13 | 3, 3 | addcld 9938 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐴) ∈ ℂ) |
14 | 13, 4, 4 | addassd 9941 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = ((𝐴 + 𝐴) + (𝐵 + 𝐵))) |
15 | 6, 3, 4 | addassd 9941 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + (𝐴 + 𝐵))) |
16 | 12, 14, 15 | 3eqtr4d 2654 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵)) |
17 | 13, 4 | addcld 9938 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) ∈ ℂ) |
18 | 6, 3 | addcld 9938 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) ∈ ℂ) |
19 | addcan2 10100 | . . . . 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 1318 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 + 𝐴) + 𝐵) + 𝐵) = (((𝐴 + 𝐵) + 𝐴) + 𝐵) ↔ ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴))) |
21 | 16, 20 | mpbid 221 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = ((𝐴 + 𝐵) + 𝐴)) |
22 | 3, 3, 4 | addassd 9941 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐴) + 𝐵) = (𝐴 + (𝐴 + 𝐵))) |
23 | 3, 4, 3 | addassd 9941 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐴) = (𝐴 + (𝐵 + 𝐴))) |
24 | 21, 22, 23 | 3eqtr3d 2652 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴))) |
25 | 4, 3 | addcld 9938 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) ∈ ℂ) |
26 | addcan 10099 | . . 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 1318 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (𝐴 + 𝐵)) = (𝐴 + (𝐵 + 𝐴)) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
28 | 24, 27 | mpbid 221 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: addcomi 10106 ltaddnegr 10131 add12 10132 add32 10133 add42 10136 subsub23 10165 pncan2 10167 addsub 10171 addsub12 10173 addsubeq4 10175 sub32 10194 pnpcan2 10200 ppncan 10202 sub4 10205 negsubdi2 10219 ltaddsub2 10382 leaddsub2 10384 leltadd 10391 ltaddpos2 10398 addge02 10418 conjmul 10621 recp1lt1 10800 recreclt 10801 avgle1 11149 avgle2 11150 avgle 11151 nn0nnaddcl 11201 xaddcom 11945 fzen 12229 fzshftral 12297 fzo0addelr 12390 elfzoext 12392 flzadd 12489 addmodidr 12581 modadd2mod 12582 nn0ennn 12640 seradd 12705 bernneq2 12853 hashfz 13074 ccatalpha 13228 revccat 13366 2cshwcom 13413 shftval2 13663 shftval4 13665 crim 13703 absmax 13917 climshft2 14161 summolem3 14292 binom1dif 14404 isumshft 14410 arisum 14431 mertenslem1 14455 bpolydiflem 14624 addcos 14743 demoivreALT 14770 dvdsaddr 14863 sumodd 14949 divalglem4 14957 divalgb 14965 gcdaddm 15084 hashdvds 15318 phiprmpw 15319 pythagtriplem2 15360 prmgaplem7 15599 mulgnndir 17392 mulgnndirOLD 17393 cnaddablx 18094 cnaddabl 18095 zaddablx 18098 cncrng 19586 ioo2bl 22404 icopnfcnv 22549 uniioombllem3 23159 fta1glem1 23729 plyremlem 23863 fta1lem 23866 vieta1lem1 23869 vieta1lem2 23870 aaliou3lem2 23902 dvradcnv 23979 pserdv2 23988 reeff1olem 24004 ptolemy 24052 logcnlem4 24191 cxpsqrt 24249 atandm2 24404 atandm4 24406 atanlogsublem 24442 2efiatan 24445 dvatan 24462 birthdaylem2 24479 emcllem2 24523 fsumharmonic 24538 wilthlem1 24594 wilthlem2 24595 basellem8 24614 1sgmprm 24724 perfectlem2 24755 pntibndlem1 25078 pntibndlem2 25080 pntlemd 25083 pntlemc 25084 cnaddabloOLD 26820 cdj3lem3b 28683 isarchi3 29072 archiabllem2c 29080 cos2h 32570 tan2h 32571 eldioph2lem1 36341 addcomgi 37681 epoo 40150 perfectALTVlem2 40165 sgoldbaltlem2 40202 fz0addcom 40354 eucrctshift 41411 |
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