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Theorem mullocpr 7485

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 7389	. . . . . . . 8
2		prmuloc 7480	. . . . . . . 8 $/\$ $/\$ $/\$
3	1, 2	sylan 281	. . . . . . 7 $/\$ $/\$ $/\$
4		r2ex 2477	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	sylib 121	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6	5	adantlr 469	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	adantlr 469	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simprr3 1032	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		mulclnq 7290	. . . . . . . . 9 $/\$
11	9, 10	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		appdivnq 7477	. . . . . . . 8 $/\$ $/\$
13	8, 11, 12	syl2anc 409	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simprrr 530	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	11	adantr 274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		appdivnq 7477	. . . . . . . . 9 $/\$ $/\$
17	14, 15, 16	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simplll 523	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	ad2antrr 480	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		simprl 521	. . . . . . . . . 10 $/\$ $/\$
21	20	ad2antlr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		simprrl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		simprrr 530	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		simpllr 524	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	24	ad2antrr 480	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	9	ad2antrr 480	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		3simpa 979	. . . . . . . . . . 11 $/\$ $/\$ $/\$
28	27	ad2antll 483	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ad2antrr 480	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simplrl 525	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simprl 521	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	30, 31	jca 304	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	19, 21, 22, 23, 25, 26, 29, 32	mullocprlem 7484	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
34	17, 33	rexlimddv 2579	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	13, 34	rexlimddv 2579	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36	35	ex 114	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
37	36	exlimdvv 1877	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
38	7, 37	mpd 13	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
39	38	ex 114	. 2 $/\$ $/\$ $/\$ $\/$
40	39	ralrimivva 2539	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $\/$ wo 698 $/\$ w3a 963

wex 1472

wcel 2128

wral 2435

wrex 2436

cop 3563 class class class wbr 3965

cfv 5169 (class class class)co 5821

c1st 6083

c2nd 6084

cnq 7194

cmq 7197

cltq 7199

cnp 7205

cmp 7208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-1o 6360 df-2o 6361 df-oadd 6364 df-omul 6365 df-er 6477 df-ec 6479 df-qs 6483 df-ni 7218 df-pli 7219 df-mi 7220 df-lti 7221 df-plpq 7258 df-mpq 7259 df-enq 7261 df-nqqs 7262 df-plqqs 7263 df-mqqs 7264 df-1nqqs 7265 df-rq 7266 df-ltnqqs 7267 df-enq0 7338 df-nq0 7339 df-0nq0 7340 df-plq0 7341 df-mq0 7342 df-inp 7380 df-imp 7383

This theorem is referenced by: mulclpr 7486

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