divmulasscomap - Intuitionistic Logic Explorer

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Theorem divmulasscomap 8831

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 8830	. 2 $/\$ $/\$ $/\$ $/\$ #
2		mulcom 8116	. . . . 5 $/\$
3	2	3adant3 1041	. . . 4 $/\$ $/\$
4	3	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ #
5	4	oveq1d 6009	. 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 8813	. . . . 5 $/\$ $/\$ #
13	11, 12	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ #
14	6, 7, 13	mulassd 8158	. . 3 $/\$ $/\$ $/\$ $/\$ #
15	8	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ #
16		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ #
17		divassap 8825	. . . . . 6 $/\$ $/\$ $/\$ #
18	7, 15, 16, 17	syl3anc 1271	. . . . 5 $/\$ $/\$ $/\$ $/\$ #
19	18	eqcomd 2235	. . . 4 $/\$ $/\$ $/\$ $/\$ #
20	19	oveq2d 6010	. . 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 4082 (class class class)co 5994

cc 7985

cc0 7987

cmul 7992 # cap 8716

cdiv 8807

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 4201 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105

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 3888 df-br 4083 df-opab 4145 df-id 4381 df-po 4384 df-iso 4385 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-iota 5274 df-fun 5316 df-fv 5322 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808

This theorem is referenced by: cncongr2 12612