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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 10988 | . 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵) |
4 | addcomli.2 | . 2 ⊢ (𝐴 + 𝐵) = 𝐶 | |
5 | 3, 4 | eqtri 2759 | 1 ⊢ (𝐵 + 𝐴) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 + caddc 10697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 |
This theorem is referenced by: mvlladdi 11061 negsubdi2i 11129 1p2e3ALT 11939 4t4e16 12357 6t3e18 12363 6t5e30 12365 7t3e21 12368 7t4e28 12369 7t6e42 12371 7t7e49 12372 8t3e24 12374 8t4e32 12375 8t5e40 12376 8t8e64 12379 9t3e27 12381 9t4e36 12382 9t5e45 12383 9t6e54 12384 9t7e63 12385 9t8e72 12386 9t9e81 12387 n2dvdsm1 15893 bitsfzo 15957 gcdaddmlem 16046 6gcd4e2 16061 gcdi 16589 2exp8 16605 2exp16 16607 37prm 16637 43prm 16638 83prm 16639 139prm 16640 163prm 16641 317prm 16642 631prm 16643 1259lem1 16647 1259lem2 16648 1259lem3 16649 1259lem4 16650 1259lem5 16651 1259prm 16652 2503lem1 16653 2503lem2 16654 2503lem3 16655 2503prm 16656 4001lem1 16657 4001lem2 16658 4001lem4 16660 4001prm 16661 iaa 25172 dvradcnv 25267 eulerid 25318 binom4 25687 log2ublem3 25785 log2ub 25786 lgsdir2lem1 26160 m1lgs 26223 2lgsoddprmlem3d 26248 addsqnreup 26278 ex-exp 28487 ex-bc 28489 ex-gcd 28494 ex-ind-dvds 28498 9p10ne21 28507 vcm 28611 fib5 32038 fib6 32039 hgt750lem 32297 hgt750lem2 32298 60gcd7e1 39696 3exp7 39744 3lexlogpow5ineq1 39745 3lexlogpow5ineq5 39751 aks4d1p1p4 39761 aks4d1p1p5 39765 aks4d1p1 39766 decpmulnc 39963 sqdeccom12 39965 sq3deccom12 39966 235t711 39967 ex-decpmul 39968 resqrtvalex 40870 imsqrtvalex 40871 inductionexd 41383 lhe4.4ex1a 41561 dirkertrigeqlem1 43257 sqwvfoura 43387 sqwvfourb 43388 fourierswlem 43389 fouriersw 43390 fmtno5lem4 44624 257prm 44629 fmtno4nprmfac193 44642 fmtno5faclem3 44649 fmtno5fac 44650 139prmALT 44664 127prm 44667 11t31e341 44800 gbpart8 44836 ackval3 45645 ackval2012 45653 ackval3012 45654 |
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