divmulasscomap - Intuitionistic Logic Explorer

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Theorem divmulasscomap 8991

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 8990	. 2 $/\$ $/\$ $/\$ $/\$ #
2		mulcom 8273	. . . . 5 $/\$
3	2	3adant3 1044	. . . 4 $/\$ $/\$
4	3	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ #
5	4	oveq1d 6074	. 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 8973	. . . . 5 $/\$ $/\$ #
13	11, 12	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ #
14	6, 7, 13	mulassd 8314	. . 3 $/\$ $/\$ $/\$ $/\$ #
15	8	adantr 276	. . . . . 6 $/\$ $/\$ $/\$ $/\$ #
16		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ # $/\$ #
17		divassap 8985	. . . . . 6 $/\$ $/\$ $/\$ #
18	7, 15, 16, 17	syl3anc 1274	. . . . 5 $/\$ $/\$ $/\$ $/\$ #
19	18	eqcomd 2240	. . . 4 $/\$ $/\$ $/\$ $/\$ #
20	19	oveq2d 6075	. . 3 $/\$ $/\$ $/\$ $/\$ #
21	14, 20	eqtrd 2267	. 2 $/\$ $/\$ $/\$ $/\$ #
22	1, 5, 21	3eqtrd 2271	1 $/\$ $/\$ $/\$ $/\$ #

Colors of variables: wff set class

Syntax hints:

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

wceq 1398

wcel 2205 class class class wbr 4115 (class class class)co 6059

cc 8142

cc0 8144

cmul 8149 # cap 8874

cdiv 8967

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4234 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4665 ax-cnex 8235 ax-resscn 8236 ax-1cn 8237 ax-1re 8238 ax-icn 8239 ax-addcl 8240 ax-addrcl 8241 ax-mulcl 8242 ax-mulrcl 8243 ax-addcom 8244 ax-mulcom 8245 ax-addass 8246 ax-mulass 8247 ax-distr 8248 ax-i2m1 8249 ax-0lt1 8250 ax-1rid 8251 ax-0id 8252 ax-rnegex 8253 ax-precex 8254 ax-cnre 8255 ax-pre-ltirr 8256 ax-pre-ltwlin 8257 ax-pre-lttrn 8258 ax-pre-apti 8259 ax-pre-ltadd 8260 ax-pre-mulgt0 8261 ax-pre-mulext 8262

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-br 4116 df-opab 4178 df-id 4420 df-po 4423 df-iso 4424 df-xp 4761 df-rel 4762 df-cnv 4763 df-co 4764 df-dm 4765 df-iota 5318 df-fun 5360 df-fv 5366 df-riota 6012 df-ov 6062 df-oprab 6063 df-mpo 6064 df-pnf 8327 df-mnf 8328 df-xr 8329 df-ltxr 8330 df-le 8331 df-sub 8464 df-neg 8465 df-reap 8868 df-ap 8875 df-div 8968

This theorem is referenced by: cncongr2 12831