mul4 - Intuitionistic Logic Explorer

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Theorem mul4 7145

Description: Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

**Assertion**
Ref	Expression
mul4	$/\$ $/\$ $/\$

**Proof of Theorem mul4**
Step	Hyp	Ref	Expression
1		mul32 7143	. . . . 5 $/\$ $/\$
2	1	oveq1d 5527	. . . 4 $/\$ $/\$
3	2	3expa 1104	. . 3 $/\$ $/\$
4	3	adantrr 448	. 2 $/\$ $/\$ $/\$
5		mulcl 7008	. . 3 $/\$
6		mulass 7012	. . . 4 $/\$ $/\$
7	6	3expb 1105	. . 3 $/\$ $/\$
8	5, 7	sylan 267	. 2 $/\$ $/\$ $/\$
9		mulcl 7008	. . . 4 $/\$
10		mulass 7012	. . . . 5 $/\$ $/\$
11	10	3expb 1105	. . . 4 $/\$ $/\$
12	9, 11	sylan 267	. . 3 $/\$ $/\$ $/\$
13	12	an4s 522	. 2 $/\$ $/\$ $/\$
14	4, 8, 13	3eqtr3d 2080	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 97 $/\$ w3a 885

wceq 1243

wcel 1393 (class class class)co 5512

cc 6887

cmul 6894

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-mulcl 6982 ax-mulcom 6985 ax-mulass 6987

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515

This theorem is referenced by: mul4i 7161 mul4d 7168 recextlem1 7632 divmuldivap 7688 mulexp 9294