divmulasscomap - Intuitionistic Logic Explorer

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Theorem divmulasscomap 8859

Description: An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)

**Assertion**
Ref	Expression
divmulasscomap	$/\$ $/\$ $/\$ $/\$ #

**Proof of Theorem divmulasscomap**
Step	Hyp	Ref	Expression
1		divmulassap 8858	. 2 $/\$ $/\$ $/\$ $/\$ #
2		mulcom 8144	. . . . 5 $/\$
3	2	3adant3 1041	. . . 4 $/\$ $/\$
4	3	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ #
5	4	oveq1d 6025	. 2 $/\$ $/\$ $/\$ $/\$ #
6		simpl2 1025	. . . 4 $/\$ $/\$ $/\$ $/\$ #
7		simpl1 1024	. . . 4 $/\$ $/\$ $/\$ $/\$ #
8		simp3 1023	. . . . . . 7 $/\$ $/\$
9	8	anim1i 340	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$ #
10		3anass 1006	. . . . . 6 $/\$ $/\$ # $/\$ $/\$ #
11	9, 10	sylibr 134	. . . . 5 $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$ #
12		divclap 8841	. . . . 5 $/\$ $/\$ #
13	11, 12	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ #
14	6, 7, 13	mulassd 8186	. . 3 $/\$ $/\$ $/\$ $/\$ #
15	8	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ #
16		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ #
17		divassap 8853	. . . . . 6 $/\$ $/\$ $/\$ #
18	7, 15, 16, 17	syl3anc 1271	. . . . 5 $/\$ $/\$ $/\$ $/\$ #
19	18	eqcomd 2235	. . . 4 $/\$ $/\$ $/\$ $/\$ #
20	19	oveq2d 6026	. . 3 $/\$ $/\$ $/\$ $/\$ #
21	14, 20	eqtrd 2262	. 2 $/\$ $/\$ $/\$ $/\$ #
22	1, 5, 21	3eqtrd 2266	1 $/\$ $/\$ $/\$ $/\$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1002

wceq 1395

wcel 2200 class class class wbr 4083 (class class class)co 6010

cc 8013

cc0 8015

cmul 8020 # cap 8744

cdiv 8835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4385 df-po 4388 df-iso 4389 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-iota 5281 df-fun 5323 df-fv 5329 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836

This theorem is referenced by: cncongr2 12647