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Mirrors > Home > ILE Home > Th. List > reapmul1 | Unicode version |
Description: Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8516. (Contributed by Jim Kingdon, 8-Feb-2020.) |
Ref | Expression |
---|---|
reapmul1 | # # # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7734 | . . . . 5 | |
2 | reaplt 8318 | . . . . 5 # | |
3 | 1, 2 | mpan2 421 | . . . 4 # |
4 | 3 | pm5.32i 449 | . . 3 # |
5 | simp1 966 | . . . . . . . . . . 11 | |
6 | 5 | recnd 7762 | . . . . . . . . . 10 |
7 | simp3l 994 | . . . . . . . . . . 11 | |
8 | 7 | recnd 7762 | . . . . . . . . . 10 |
9 | 6, 8 | mulneg2d 8142 | . . . . . . . . 9 |
10 | simp2 967 | . . . . . . . . . . 11 | |
11 | 10 | recnd 7762 | . . . . . . . . . 10 |
12 | 11, 8 | mulneg2d 8142 | . . . . . . . . 9 |
13 | 9, 12 | breq12d 3912 | . . . . . . . 8 # # |
14 | 7 | renegcld 8110 | . . . . . . . . 9 |
15 | simp3r 995 | . . . . . . . . . 10 | |
16 | 7 | lt0neg1d 8245 | . . . . . . . . . 10 |
17 | 15, 16 | mpbid 146 | . . . . . . . . 9 |
18 | reapmul1lem 8324 | . . . . . . . . 9 # # | |
19 | 5, 10, 14, 17, 18 | syl112anc 1205 | . . . . . . . 8 # # |
20 | 5, 7 | remulcld 7764 | . . . . . . . . . . 11 |
21 | 10, 7 | remulcld 7764 | . . . . . . . . . . 11 |
22 | 20, 21 | ltnegd 8253 | . . . . . . . . . 10 |
23 | 21, 20 | ltnegd 8253 | . . . . . . . . . 10 |
24 | 22, 23 | orbi12d 767 | . . . . . . . . 9 |
25 | reaplt 8318 | . . . . . . . . . 10 # | |
26 | 20, 21, 25 | syl2anc 408 | . . . . . . . . 9 # |
27 | 20 | renegcld 8110 | . . . . . . . . . . 11 |
28 | 21 | renegcld 8110 | . . . . . . . . . . 11 |
29 | reaplt 8318 | . . . . . . . . . . 11 # | |
30 | 27, 28, 29 | syl2anc 408 | . . . . . . . . . 10 # |
31 | orcom 702 | . . . . . . . . . 10 | |
32 | 30, 31 | syl6bb 195 | . . . . . . . . 9 # |
33 | 24, 26, 32 | 3bitr4d 219 | . . . . . . . 8 # # |
34 | 13, 19, 33 | 3bitr4d 219 | . . . . . . 7 # # |
35 | 34 | 3expa 1166 | . . . . . 6 # # |
36 | 35 | anassrs 397 | . . . . 5 # # |
37 | reapmul1lem 8324 | . . . . . . 7 # # | |
38 | 37 | 3expa 1166 | . . . . . 6 # # |
39 | 38 | anassrs 397 | . . . . 5 # # |
40 | 36, 39 | jaodan 771 | . . . 4 # # |
41 | 40 | anasss 396 | . . 3 # # |
42 | 4, 41 | sylan2b 285 | . 2 # # # |
43 | 42 | 3impa 1161 | 1 # # # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 w3a 947 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cmul 7593 clt 7768 cneg 7902 # cap 8311 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 |
This theorem is referenced by: (None) |
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