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| Mirrors > Home > MPE Home > Th. List > addcomi | Structured version Visualization version GIF version | ||
| Description: Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
| Ref | Expression |
|---|---|
| mul.1 | ⊢ 𝐴 ∈ ℂ |
| mul.2 | ⊢ 𝐵 ∈ ℂ |
| Ref | Expression |
|---|---|
| addcomi | ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
| 2 | mul.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
| 3 | addcom 11447 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 + caddc 11158 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: addcomli 11453 comraddi 11476 fztpval 13626 fzo1to4tp 13793 ef01bndlem 16220 modxai 17106 pcoass 25057 tangtx 26547 eff1o 26591 log2ublem2 26990 basellem9 27132 ppiub 27248 bposlem8 27335 lgsdir2lem2 27370 lgsdir2lem3 27371 lgsdir2lem5 27373 ax5seglem7 28950 ipasslem10 30858 normlem2 31130 normlem3 31131 norm-ii-i 31156 normpar2i 31175 dpmul4 32896 hgt750lem2 34667 problem3 35672 problem5 35674 quad3 35675 mblfinlem3 37666 fdc 37752 addcomnni 41986 gcdaddmzz2nncomi 41996 aks4d1p1p4 42072 2xp3dxp2ge1d 42242 decaddcom 42319 sqdeccom12 42324 stoweidlem13 46028 fourierdlem24 46146 3exp4mod41 47603 |
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