muladd - Intuitionistic Logic Explorer

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Theorem muladd 7923

Description: Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

**Assertion**
Ref	Expression
muladd	$/\$ $/\$ $/\$

**Proof of Theorem muladd**
Step	Hyp	Ref	Expression
1		addcl 7528	. . 3 $/\$
2		adddi 7535	. . . 4 $/\$ $/\$
3	2	3expb 1145	. . 3 $/\$ $/\$
4	1, 3	sylan 278	. 2 $/\$ $/\$ $/\$
5		adddir 7540	. . . . 5 $/\$ $/\$
6	5	3expa 1144	. . . 4 $/\$ $/\$
7	6	adantrr 464	. . 3 $/\$ $/\$ $/\$
8		adddir 7540	. . . . 5 $/\$ $/\$
9	8	3expa 1144	. . . 4 $/\$ $/\$
10	9	adantrl 463	. . 3 $/\$ $/\$ $/\$
11	7, 10	oveq12d 5684	. 2 $/\$ $/\$ $/\$
12		mulcl 7530	. . . . 5 $/\$
13	12	ad2ant2r 494	. . . 4 $/\$ $/\$ $/\$
14		mulcl 7530	. . . . 5 $/\$
15	14	ad2ant2lr 495	. . . 4 $/\$ $/\$ $/\$
16		mulcl 7530	. . . . . . 7 $/\$
17		mulcl 7530	. . . . . . 7 $/\$
18		addcl 7528	. . . . . . 7 $/\$
19	16, 17, 18	syl2an 284	. . . . . 6 $/\$ $/\$ $/\$
20	19	anandirs 561	. . . . 5 $/\$ $/\$
21	20	adantrl 463	. . . 4 $/\$ $/\$ $/\$
22	13, 15, 21	add32d 7711	. . 3 $/\$ $/\$ $/\$
23		mulcom 7532	. . . . . . 7 $/\$
24	23	ad2ant2l 493	. . . . . 6 $/\$ $/\$ $/\$
25	24	oveq2d 5682	. . . . 5 $/\$ $/\$ $/\$
26	16	ad2ant2rl 496	. . . . . 6 $/\$ $/\$ $/\$
27	17	ad2ant2l 493	. . . . . 6 $/\$ $/\$ $/\$
28	13, 26, 27	addassd 7571	. . . . 5 $/\$ $/\$ $/\$
29		mulcl 7530	. . . . . . . 8 $/\$
30	29	ancoms 265	. . . . . . 7 $/\$
31	30	ad2ant2l 493	. . . . . 6 $/\$ $/\$ $/\$
32	13, 26, 31	add32d 7711	. . . . 5 $/\$ $/\$ $/\$
33	25, 28, 32	3eqtr3d 2129	. . . 4 $/\$ $/\$ $/\$
34		mulcom 7532	. . . . 5 $/\$
35	34	ad2ant2lr 495	. . . 4 $/\$ $/\$ $/\$
36	33, 35	oveq12d 5684	. . 3 $/\$ $/\$ $/\$
37		addcl 7528	. . . . . 6 $/\$
38	12, 30, 37	syl2an 284	. . . . 5 $/\$ $/\$ $/\$
39	38	an4s 556	. . . 4 $/\$ $/\$ $/\$
40		mulcl 7530	. . . . . 6 $/\$
41	40	ancoms 265	. . . . 5 $/\$
42	41	ad2ant2lr 495	. . . 4 $/\$ $/\$ $/\$
43	39, 26, 42	addassd 7571	. . 3 $/\$ $/\$ $/\$
44	22, 36, 43	3eqtrd 2125	. 2 $/\$ $/\$ $/\$
45	4, 11, 44	3eqtrd 2125	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1290

wcel 1439 (class class class)co 5666

cc 7409

caddc 7414

cmul 7416

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-addcl 7502 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-distr 7510

This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rex 2366 df-v 2622 df-un 3004 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669

This theorem is referenced by: mulsub 7940 muladdi 7948 muladdd 7955 sqabsadd 10549 demoivreALT 11124

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