mullocpr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mullocpr		Unicode version

Theorem mullocpr 7719

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 7623	. . . . . . . 8
2		prmuloc 7714	. . . . . . . 8 $/\$ $/\$ $/\$
3	1, 2	sylan 283	. . . . . . 7 $/\$ $/\$ $/\$
4		r2ex 2528	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	sylib 122	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
6	5	adantlr 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	adantlr 477	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simprr3 1050	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		mulclnq 7524	. . . . . . . . 9 $/\$
11	9, 10	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		appdivnq 7711	. . . . . . . 8 $/\$ $/\$
13	8, 11, 12	syl2anc 411	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simprrr 540	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	11	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		appdivnq 7711	. . . . . . . . 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 997	. . . . . . . . . . 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 7718	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
34	17, 33	rexlimddv 2630	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
35	13, 34	rexlimddv 2630	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
36	35	ex 115	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
37	36	exlimdvv 1922	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
38	7, 37	mpd 13	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
39	38	ex 115	. 2 $/\$ $/\$ $/\$ $\/$
40	39	ralrimivva 2590	1 $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 710 $/\$ w3a 981

wex 1516

wcel 2178

wral 2486

wrex 2487

cop 3646 class class class wbr 4059

cfv 5290 (class class class)co 5967

c1st 6247

c2nd 6248

cnq 7428

cmq 7431

cltq 7433

cnp 7439

cmp 7442

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-eprel 4354 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-1o 6525 df-2o 6526 df-oadd 6529 df-omul 6530 df-er 6643 df-ec 6645 df-qs 6649 df-ni 7452 df-pli 7453 df-mi 7454 df-lti 7455 df-plpq 7492 df-mpq 7493 df-enq 7495 df-nqqs 7496 df-plqqs 7497 df-mqqs 7498 df-1nqqs 7499 df-rq 7500 df-ltnqqs 7501 df-enq0 7572 df-nq0 7573 df-0nq0 7574 df-plq0 7575 df-mq0 7576 df-inp 7614 df-imp 7617

This theorem is referenced by: mulclpr 7720

W3C validator