mul12 - Metamath Proof Explorer

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Theorem mul12 10202

Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

**Assertion**
Ref	Expression
mul12	$/\$ $/\$

**Proof of Theorem mul12**
Step	Hyp	Ref	Expression
1		mulcom 10022	. . . 4 $/\$
2	1	oveq1d 6665	. . 3 $/\$
3	2	3adant3 1081	. 2 $/\$ $/\$
4		mulass 10024	. 2 $/\$ $/\$
5		mulass 10024	. . 3 $/\$ $/\$
6	5	3com12 1269	. 2 $/\$ $/\$
7	3, 4, 6	3eqtr3d 2664	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990 (class class class)co 6650

cc 9934

cmul 9941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-mulcom 10000 ax-mulass 10002

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653

This theorem is referenced by: mul02 10214 mul12i 10231 mul12d 10245 mulre 13861 sqreulem 14099 fsumcube 14791 demoivre 14930 demoivreALT 14931 dvdscmul 15008 dvdscmulr 15010 dvdstr 15018 ablfacrp 18465 nmoleub2lem3 22915 sinperlem 24232 coskpi 24272 sineq0 24273 efif1olem4 24291 rpvmasum2 25201 expgrowthi 38532