rexmul - Metamath Proof Explorer

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Theorem rexmul 12101

Description: The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
rexmul	$/\$

**Proof of Theorem rexmul**
Step	Hyp	Ref	Expression
1		renepnf 10087	. . . . . . . . . . 11
2	1	adantr 481	. . . . . . . . . 10 $/\$
3	2	necon2bi 2824	. . . . . . . . 9 $/\$
4	3	adantl 482	. . . . . . . 8 $/\$ $/\$
5		renemnf 10088	. . . . . . . . . . 11
6	5	adantr 481	. . . . . . . . . 10 $/\$
7	6	necon2bi 2824	. . . . . . . . 9 $/\$
8	7	adantl 482	. . . . . . . 8 $/\$ $/\$
9	4, 8	jaoi 394	. . . . . . 7 $/\$ $\/$ $/\$ $/\$
10		renepnf 10087	. . . . . . . . . . 11
11	10	adantl 482	. . . . . . . . . 10 $/\$
12	11	necon2bi 2824	. . . . . . . . 9 $/\$
13	12	adantl 482	. . . . . . . 8 $/\$ $/\$
14		renemnf 10088	. . . . . . . . . . 11
15	14	adantl 482	. . . . . . . . . 10 $/\$
16	15	necon2bi 2824	. . . . . . . . 9 $/\$
17	16	adantl 482	. . . . . . . 8 $/\$ $/\$
18	13, 17	jaoi 394	. . . . . . 7 $/\$ $\/$ $/\$ $/\$
19	9, 18	jaoi 394	. . . . . 6 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$
20	19	con2i 134	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
21	20	iffalsed 4097	. . . 4 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
22	7	adantl 482	. . . . . . . 8 $/\$ $/\$
23	3	adantl 482	. . . . . . . 8 $/\$ $/\$
24	22, 23	jaoi 394	. . . . . . 7 $/\$ $\/$ $/\$ $/\$
25	16	adantl 482	. . . . . . . 8 $/\$ $/\$
26	12	adantl 482	. . . . . . . 8 $/\$ $/\$
27	25, 26	jaoi 394	. . . . . . 7 $/\$ $\/$ $/\$ $/\$
28	24, 27	jaoi 394	. . . . . 6 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$
29	28	con2i 134	. . . . 5 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
30	29	iffalsed 4097	. . . 4 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
31	21, 30	eqtrd 2656	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
32	31	ifeq2d 4105	. 2 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$
33		rexr 10085	. . 3
34		rexr 10085	. . 3
35		xmulval 12056	. . 3 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
36	33, 34, 35	syl2an 494	. 2 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
37		ifid 4125	. . 3 $\/$
38		oveq1 6657	. . . . . 6
39		mul02lem2 10213	. . . . . . 7
40	39	adantl 482	. . . . . 6 $/\$
41	38, 40	sylan9eqr 2678	. . . . 5 $/\$ $/\$
42		oveq2 6658	. . . . . 6
43		recn 10026	. . . . . . . 8
44	43	mul01d 10235	. . . . . . 7
45	44	adantr 481	. . . . . 6 $/\$
46	42, 45	sylan9eqr 2678	. . . . 5 $/\$ $/\$
47	41, 46	jaodan 826	. . . 4 $/\$ $/\$ $\/$
48	47	ifeq1da 4116	. . 3 $/\$ $\/$ $\/$
49	37, 48	syl5eqr 2670	. 2 $/\$ $\/$
50	32, 36, 49	3eqtr4d 2666	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

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

wceq 1483

wcel 1990

wne 2794

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

cr 9935

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-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-xmul 11948

This theorem is referenced by: xmulid1 12109 xmulgt0 12113 xmulasslem3 12116 xlemul1a 12118 xlemul1 12120 xadddilem 12124 nmoix 22533 nmoi2 22534 metnrmlem3 22664 nmoleub2lem 22914 xrecex 29628 rexdiv 29634 pnfinf 29737 xrge0slmod 29844 esumcst 30125 omssubadd 30362

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