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Theorem mullocpr 7758

Description: Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both

and

are positive, not just

). (Contributed by Jim Kingdon, 8-Dec-2019.)

**Assertion**
Ref	Expression
mullocpr	$/\$ $\/$

Distinct variable groups:

Proof of Theorem mullocpr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7662	. . . . . . . 8
2		prmuloc 7753	. . . . . . . 8 $/\$ $/\$ $/\$
3	1, 2	sylan 283	. . . . . . 7 $/\$ $/\$ $/\$
4		r2ex 2550	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	sylib 122	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6	5	adantlr 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	adantlr 477	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simprr3 1071	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		mulclnq 7563	. . . . . . . . 9 $/\$
11	9, 10	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		appdivnq 7750	. . . . . . . 8 $/\$ $/\$
13	8, 11, 12	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simprrr 540	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		appdivnq 7750	. . . . . . . . 9 $/\$ $/\$
17	14, 15, 16	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simplll 533	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 529	. . . . . . . . . 10 $/\$ $/\$
21	20	ad2antlr 489	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simprrl 539	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprrr 540	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		simpllr 534	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	24	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	9	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		3simpa 1018	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	ad2antll 491	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ad2antrr 488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simplrl 535	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simprl 529	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	30, 31	jca 306	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	19, 21, 22, 23, 25, 26, 29, 32	mullocprlem 7757	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
34	17, 33	rexlimddv 2653	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	13, 34	rexlimddv 2653	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36	35	ex 115	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
37	36	exlimdvv 1944	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
38	7, 37	mpd 13	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
39	38	ex 115	. 2 $/\$ $/\$ $/\$ $\/$
40	39	ralrimivva 2612	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 713 $/\$ w3a 1002

wex 1538

wcel 2200

wral 2508

wrex 2509

cop 3669 class class class wbr 4083

cfv 5318 (class class class)co 6001

c1st 6284

c2nd 6285

cnq 7467

cmq 7470

cltq 7472

cnp 7478

cmp 7481

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-irdg 6516 df-1o 6562 df-2o 6563 df-oadd 6566 df-omul 6567 df-er 6680 df-ec 6682 df-qs 6686 df-ni 7491 df-pli 7492 df-mi 7493 df-lti 7494 df-plpq 7531 df-mpq 7532 df-enq 7534 df-nqqs 7535 df-plqqs 7536 df-mqqs 7537 df-1nqqs 7538 df-rq 7539 df-ltnqqs 7540 df-enq0 7611 df-nq0 7612 df-0nq0 7613 df-plq0 7614 df-mq0 7615 df-inp 7653 df-imp 7656

This theorem is referenced by: mulclpr 7759

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