divmulasscomap - Intuitionistic Logic Explorer

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Theorem divmulasscomap 8969

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 8968	. 2 $/\$ $/\$ $/\$ $/\$ #
2		mulcom 8255	. . . . 5 $/\$
3	2	3adant3 1044	. . . 4 $/\$ $/\$
4	3	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ #
5	4	oveq1d 6064	. 2 $/\$ $/\$ $/\$ $/\$ #
6		simpl2 1028	. . . 4 $/\$ $/\$ $/\$ $/\$ #
7		simpl1 1027	. . . 4 $/\$ $/\$ $/\$ $/\$ #
8		simp3 1026	. . . . . . 7 $/\$ $/\$
9	8	anim1i 340	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$ #
10		3anass 1009	. . . . . 6 $/\$ $/\$ # $/\$ $/\$ #
11	9, 10	sylibr 134	. . . . 5 $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$ #
12		divclap 8951	. . . . 5 $/\$ $/\$ #
13	11, 12	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ #
14	6, 7, 13	mulassd 8296	. . 3 $/\$ $/\$ $/\$ $/\$ #
15	8	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ #
16		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ #
17		divassap 8963	. . . . . 6 $/\$ $/\$ $/\$ #
18	7, 15, 16, 17	syl3anc 1274	. . . . 5 $/\$ $/\$ $/\$ $/\$ #
19	18	eqcomd 2238	. . . 4 $/\$ $/\$ $/\$ $/\$ #
20	19	oveq2d 6065	. . 3 $/\$ $/\$ $/\$ $/\$ #
21	14, 20	eqtrd 2265	. 2 $/\$ $/\$ $/\$ $/\$ #
22	1, 5, 21	3eqtrd 2269	1 $/\$ $/\$ $/\$ $/\$ #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1005

wceq 1398

wcel 2203 class class class wbr 4108 (class class class)co 6049

cc 8124

cc0 8126

cmul 8131 # cap 8854

cdiv 8945

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-br 4109 df-opab 4171 df-id 4413 df-po 4416 df-iso 4417 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-iota 5311 df-fun 5353 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946

This theorem is referenced by: cncongr2 12797