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| Mirrors > Home > ILE Home > Th. List > divmuleqap | Unicode version | ||
| Description: Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| Ref | Expression |
|---|---|
| divmuleqap |
     ![/\](wedge.gif "/\"){kind=small}    #  
#      |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | divclap 8345 |
. . . . 5
# | |
| 2 | 1 | 3expb 1163 |
. . . 4
# |
| 3 | 2 | ad2ant2r 498 |
. . 3
#
# |
| 4 | divclap 8345 |
. . . . 5
# | |
| 5 | 4 | 3expb 1163 |
. . . 4
# |
| 6 | 5 | ad2ant2l 497 |
. . 3
#
# |
| 7 | mulcl 7665 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 498 |
. . . . 5
#
# |
| 9 | mulap0 8322 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 302 |
. . . 4
#
# # |
| 11 | 10 | adantl 273 |
. . 3
#
# # |
| 12 | mulcanap2 8334 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1197 |
. 2
#
#
|
| 14 | simprll 509 |
. . . . 5
#
# | |
| 15 | simprrl 511 |
. . . . 5
#
# | |
| 16 | 3, 14, 15 | mulassd 7707 |
. . . 4
#
# |
| 17 | divcanap1 8348 |
. . . . . . 7
# | |
| 18 | 17 | 3expb 1163 |
. . . . . 6
# |
| 19 | 18 | ad2ant2r 498 |
. . . . 5
#
# |
| 20 | 19 | oveq1d 5741 |
. . . 4
#
# |
| 21 | 16, 20 | eqtr3d 2147 |
. . 3
#
# |
| 22 | 14, 15 | mulcomd 7705 |
. . . . 5
#
# |
| 23 | 22 | oveq2d 5742 |
. . . 4
#
# |
| 24 | 6, 15, 14 | mulassd 7707 |
. . . 4
#
# |
| 25 | divcanap1 8348 |
. . . . . . 7
# | |
| 26 | 25 | 3expb 1163 |
. . . . . 6
# |
| 27 | 26 | ad2ant2l 497 |
. . . . 5
#
# |
| 28 | 27 | oveq1d 5741 |
. . . 4
#
# |
| 29 | 23, 24, 28 | 3eqtr2d 2151 |
. . 3
#
# |
| 30 | 21, 29 | eqeq12d 2127 |
. 2
#
#
|
| 31 | 13, 30 | bitr3d 189 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1312 wcel 1461 class class class wbr 3893
(class class class)co 5726 cc 7539 cc0 7541 cmul 7546 # cap 8255
cdiv 8339 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 |
| This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-id 4173 df-po 4176 df-iso 4177 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-iota 5044 df-fun 5081 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 |
| This theorem is referenced by: qtri3or 9907 |
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