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Mirrors > Home > MPE Home > Th. List > addcomli | Unicode version |
Description: Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
mul.1 |
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mul.2 |
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addcomli.2 |
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Ref | Expression |
---|---|
addcomli |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.2 |
. . 3
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2 | mul.1 |
. . 3
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3 | 1, 2 | addcomi 9221 |
. 2
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4 | addcomli.2 |
. 2
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5 | 3, 4 | eqtri 2432 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem is referenced by: negsubdi2i 9350 4t4e16 10419 6t3e18 10424 6t5e30 10426 7t3e21 10429 7t4e28 10430 7t6e42 10432 7t7e49 10433 8t3e24 10435 8t4e32 10436 8t5e40 10437 8t8e64 10440 9t3e27 10442 9t4e36 10443 9t5e45 10444 9t6e54 10445 9t7e63 10446 9t8e72 10447 9t9e81 10448 binom3 11463 bitsfzo 12910 gcdaddmlem 12991 gcdi 13372 2exp8 13386 2exp16 13387 prmlem1a 13392 23prm 13404 prmlem2 13405 37prm 13406 43prm 13407 83prm 13408 139prm 13409 163prm 13410 317prm 13411 631prm 13412 1259lem1 13413 1259lem2 13414 1259lem3 13415 1259lem4 13416 1259lem5 13417 1259prm 13418 2503lem1 13419 2503lem2 13420 2503lem3 13421 2503prm 13422 4001lem1 13423 4001lem2 13424 4001lem4 13426 4001prm 13427 iaa 20203 dvradcnv 20298 eulerid 20343 binom4 20651 quart1lem 20656 log2ublem3 20749 log2ub 20750 lgsdir2lem1 21068 m1lgs 21107 pntibndlem2 21246 1kp2ke3k 21715 vcm 22011 4bc3eq4 25164 bpoly4 26017 lhe4.4ex1a 27422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pow 4345 ax-pr 4371 ax-un 4668 ax-resscn 9011 ax-1cn 9012 ax-icn 9013 ax-addcl 9014 ax-addrcl 9015 ax-mulcl 9016 ax-mulrcl 9017 ax-mulcom 9018 ax-addass 9019 ax-mulass 9020 ax-distr 9021 ax-i2m1 9022 ax-1ne0 9023 ax-1rid 9024 ax-rnegex 9025 ax-rrecex 9026 ax-cnre 9027 ax-pre-lttri 9028 ax-pre-lttrn 9029 ax-pre-ltadd 9030 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-nel 2578 df-ral 2679 df-rex 2680 df-rab 2683 df-v 2926 df-sbc 3130 df-csb 3220 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-op 3791 df-uni 3984 df-br 4181 df-opab 4235 df-mpt 4236 df-id 4466 df-po 4471 df-so 4472 df-xp 4851 df-rel 4852 df-cnv 4853 df-co 4854 df-dm 4855 df-rn 4856 df-res 4857 df-ima 4858 df-iota 5385 df-fun 5423 df-fn 5424 df-f 5425 df-f1 5426 df-fo 5427 df-f1o 5428 df-fv 5429 df-ov 6051 df-er 6872 df-en 7077 df-dom 7078 df-sdom 7079 df-pnf 9086 df-mnf 9087 df-ltxr 9089 |
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