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Theorem xmulval 12056

Description: Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulval	$/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Proof of Theorem xmulval Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 $/\$
2	1	eqeq1d 2624	. . . 4 $/\$
3		simpr 477	. . . . 5 $/\$
4	3	eqeq1d 2624	. . . 4 $/\$
5	2, 4	orbi12d 746	. . 3 $/\$ $\/$ $\/$
6	3	breq2d 4665	. . . . . . 7 $/\$
7	1	eqeq1d 2624	. . . . . . 7 $/\$
8	6, 7	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$
9	3	breq1d 4663	. . . . . . 7 $/\$
10	1	eqeq1d 2624	. . . . . . 7 $/\$
11	9, 10	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$
12	8, 11	orbi12d 746	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
13	1	breq2d 4665	. . . . . . 7 $/\$
14	3	eqeq1d 2624	. . . . . . 7 $/\$
15	13, 14	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$
16	1	breq1d 4663	. . . . . . 7 $/\$
17	3	eqeq1d 2624	. . . . . . 7 $/\$
18	16, 17	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$
19	15, 18	orbi12d 746	. . . . 5 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
20	12, 19	orbi12d 746	. . . 4 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
21	6, 10	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$
22	9, 7	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$
23	21, 22	orbi12d 746	. . . . . 6 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
24	13, 17	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$
25	16, 14	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$
26	24, 25	orbi12d 746	. . . . . 6 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$
27	23, 26	orbi12d 746	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		oveq12 6659	. . . . 5 $/\$
29	27, 28	ifbieq2d 4111	. . . 4 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	20, 29	ifbieq2d 4111	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
31	5, 30	ifbieq2d 4111	. 2 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
32		df-xmul 11948	. 2 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33		c0ex 10034	. . 3
34		pnfex 10093	. . . 4
35		mnfxr 10096	. . . . . 6
36	35	elexi 3213	. . . . 5
37		ovex 6678	. . . . 5
38	36, 37	ifex 4156	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
39	34, 38	ifex 4156	. . 3 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
40	33, 39	ifex 4156	. 2 $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
41	31, 32, 40	ovmpt2a 6791	1 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

cif 4086 class class class wbr 4653 (class class class)co 6650

cc0 9936

cmul 9941

cpnf 10071

cmnf 10072

cxr 10073

clt 10074

cxmu 11945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pnf 10076 df-mnf 10077 df-xr 10078 df-xmul 11948

This theorem is referenced by: xmulcom 12096 xmul01 12097 xmulneg1 12099 rexmul 12101 xmulpnf1 12104

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