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Mirrors > Home > MPE Home > Th. List > mulcomli | Structured version Visualization version GIF version |
Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
axi.2 | ⊢ 𝐵 ∈ ℂ |
mulcomli.3 | ⊢ (𝐴 · 𝐵) = 𝐶 |
Ref | Expression |
---|---|
mulcomli | ⊢ (𝐵 · 𝐴) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
2 | axi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 1, 2 | mulcomi 9925 | . 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
4 | mulcomli.3 | . 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 · cmul 9820 |
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-ext 2590 ax-mulcom 9879 |
This theorem depends on definitions: df-bi 196 df-an 385 df-cleq 2603 |
This theorem is referenced by: divcan1i 10648 mvllmuli 10737 recgt0ii 10808 nummul2c 11439 halfthird 11561 5recm6rec 11562 sq4e2t8 12824 cos2bnd 14757 prmo3 15583 dec5nprm 15608 decexp2 15617 karatsuba 15630 karatsubaOLD 15631 2exp6 15633 2exp8 15634 2exp16 15635 7prm 15655 13prm 15661 17prm 15662 19prm 15663 23prm 15664 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 4001lem3 15688 4001lem4 15689 4001prm 15690 pcoass 22632 efif1olem2 24093 mcubic 24374 quart1lem 24382 quart1 24383 quartlem1 24384 tanatan 24446 log2ublem3 24475 log2ub 24476 cht3 24699 bclbnd 24805 bpos1lem 24807 bposlem4 24812 bposlem5 24813 bposlem8 24816 2lgslem3a 24921 2lgsoddprmlem3c 24937 2lgsoddprmlem3d 24938 ex-fac 26700 ex-prmo 26708 ipasslem10 27078 siii 27092 normlem3 27353 bcsiALT 27420 inductionexd 37473 fouriersw 39124 1t10e1p1e11 39937 1t10e1p1e11OLD 39938 fmtno5lem1 40003 fmtno5lem2 40004 257prm 40011 fmtno4prmfac 40022 fmtno4nprmfac193 40024 fmtno5faclem2 40030 139prmALT 40049 127prm 40053 2exp11 40055 mod42tp1mod8 40057 3exp4mod41 40071 41prothprmlem2 40073 |
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