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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 10423 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
4 | 1, 2, 3 | mp2an 664 | 1 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1630 ∈ wcel 2144 (class class class)co 6792 ℂcc 10135 + caddc 10140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-ltxr 10280 |
This theorem is referenced by: addcomli 10429 fztpval 12608 fzo0to42pr 12762 fzo1to4tp 12763 0.999...OLD 14819 ef01bndlem 15119 modxai 15978 pcoass 23042 iblitg 23754 tangtx 24477 eff1o 24515 ang180lem2 24760 log2ublem2 24894 basellem9 25035 ppiub 25149 bposlem8 25236 lgsdir2lem1 25270 lgsdir2lem2 25271 lgsdir2lem3 25272 lgsdir2lem5 25274 ax5seglem7 26035 ex-exp 27643 ipasslem10 28028 normlem2 28302 normlem3 28303 norm-ii-i 28328 normpar2i 28347 dpmul4 29956 hgt750lem2 31064 problem3 31893 problem5 31895 quad3 31896 mblfinlem3 33774 fdc 33866 stoweidlem13 40741 fourierdlem24 40859 3exp4mod41 42051 comraddi 43033 |
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