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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 10106 | . 2 ⊢ (𝐵 + 𝐴) = (𝐴 + 𝐵) |
4 | addcomli.2 | . 2 ⊢ (𝐴 + 𝐵) = 𝐶 | |
5 | 3, 4 | eqtri 2632 | 1 ⊢ (𝐵 + 𝐴) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 + caddc 9818 |
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: mvlladdi 10178 negsubdi2i 10246 1p2e3 11029 4t4e16 11509 6t3e18 11518 6t5e30 11520 6t5e30OLD 11521 7t3e21 11525 7t4e28 11526 7t6e42 11528 7t7e49 11529 8t3e24 11531 8t4e32 11532 8t5e40 11533 8t5e40OLD 11534 8t8e64 11538 9t3e27 11540 9t4e36 11541 9t5e45 11542 9t6e54 11543 9t7e63 11544 9t8e72 11545 9t9e81 11546 4bc3eq4 12977 n2dvdsm1 14943 bitsfzo 14995 gcdaddmlem 15083 6gcd4e2 15093 gcdi 15615 2exp8 15634 2exp16 15635 37prm 15666 43prm 15667 83prm 15668 139prm 15669 163prm 15670 317prm 15671 631prm 15672 1259lem1 15676 1259lem2 15677 1259lem3 15678 1259lem4 15679 1259lem5 15680 1259prm 15681 2503lem1 15682 2503lem2 15683 2503lem3 15684 2503prm 15685 4001lem1 15686 4001lem2 15687 4001lem4 15689 4001prm 15690 iaa 23884 dvradcnv 23979 eulerid 24030 binom4 24377 log2ublem3 24475 log2ub 24476 lgsdir2lem1 24850 m1lgs 24913 2lgsoddprmlem3d 24938 ex-bc 26701 ex-gcd 26706 ex-ind-dvds 26710 vcm 26815 fib5 29794 fib6 29795 inductionexd 37473 lhe4.4ex1a 37550 dirkertrigeqlem1 38991 sqwvfoura 39121 sqwvfourb 39122 fourierswlem 39123 fouriersw 39124 fmtno5lem4 40006 257prm 40011 fmtno4nprmfac193 40024 fmtno5faclem3 40031 fmtno5fac 40032 139prmALT 40049 127prm 40053 gbpart8 40190 |
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