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Mirrors > Home > MPE Home > Th. List > addcomli | Structured version Visualization version GIF version |
Description: Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
mul.1 | ⊢ 𝐴 ∈ ℂ |
mul.2 | ⊢ 𝐵 ∈ ℂ |
addcomli.2 | ⊢ (𝐴 + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
addcomli | ⊢ (𝐵 + 𝐴) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
2 | mul.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 1, 2 | addcomi 10825 | . 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵) |
4 | addcomli.2 | . 2 ⊢ (𝐴 + 𝐵) = 𝐶 | |
5 | 3, 4 | eqtri 2844 | 1 ⊢ (𝐵 + 𝐴) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 + caddc 10534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 |
This theorem is referenced by: mvlladdi 10898 negsubdi2i 10966 1p2e3ALT 11775 4t4e16 12191 6t3e18 12197 6t5e30 12199 7t3e21 12202 7t4e28 12203 7t6e42 12205 7t7e49 12206 8t3e24 12208 8t4e32 12209 8t5e40 12210 8t8e64 12213 9t3e27 12215 9t4e36 12216 9t5e45 12217 9t6e54 12218 9t7e63 12219 9t8e72 12220 9t9e81 12221 n2dvdsm1 15713 bitsfzo 15778 gcdaddmlem 15866 6gcd4e2 15880 gcdi 16403 2exp8 16417 2exp16 16418 37prm 16448 43prm 16449 83prm 16450 139prm 16451 163prm 16452 317prm 16453 631prm 16454 1259lem1 16458 1259lem2 16459 1259lem3 16460 1259lem4 16461 1259lem5 16462 1259prm 16463 2503lem1 16464 2503lem2 16465 2503lem3 16466 2503prm 16467 4001lem1 16468 4001lem2 16469 4001lem4 16471 4001prm 16472 iaa 24908 dvradcnv 25003 eulerid 25054 binom4 25422 log2ublem3 25520 log2ub 25521 lgsdir2lem1 25895 m1lgs 25958 2lgsoddprmlem3d 25983 addsqnreup 26013 ex-exp 28223 ex-bc 28225 ex-gcd 28230 ex-ind-dvds 28234 9p10ne21 28243 vcm 28347 fib5 31658 fib6 31659 hgt750lem 31917 hgt750lem2 31918 decpmulnc 39166 sqdeccom12 39168 sq3deccom12 39169 235t711 39170 ex-decpmul 39171 inductionexd 40498 lhe4.4ex1a 40654 dirkertrigeqlem1 42377 sqwvfoura 42507 sqwvfourb 42508 fourierswlem 42509 fouriersw 42510 fmtno5lem4 43712 257prm 43717 fmtno4nprmfac193 43730 fmtno5faclem3 43737 fmtno5fac 43738 139prmALT 43753 127prm 43757 11t31e341 43891 gbpart8 43927 |
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