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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 8624 |
. . . . 5
# | |
| 2 | 1 | 3expb 1204 |
. . . 4
# |
| 3 | 2 | ad2ant2r 509 |
. . 3
#
# |
| 4 | divclap 8624 |
. . . . 5
# | |
| 5 | 4 | 3expb 1204 |
. . . 4
# |
| 6 | 5 | ad2ant2l 508 |
. . 3
#
# |
| 7 | mulcl 7929 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 509 |
. . . . 5
#
# |
| 9 | mulap0 8600 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | mulcanap2 8612 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1238 |
. 2
#
#
|
| 14 | simprll 537 |
. . . . 5
#
# | |
| 15 | simprrl 539 |
. . . . 5
#
# | |
| 16 | 3, 14, 15 | mulassd 7971 |
. . . 4
#
# |
| 17 | divcanap1 8627 |
. . . . . . 7
# | |
| 18 | 17 | 3expb 1204 |
. . . . . 6
# |
| 19 | 18 | ad2ant2r 509 |
. . . . 5
#
# |
| 20 | 19 | oveq1d 5884 |
. . . 4
#
# |
| 21 | 16, 20 | eqtr3d 2212 |
. . 3
#
# |
| 22 | 14, 15 | mulcomd 7969 |
. . . . 5
#
# |
| 23 | 22 | oveq2d 5885 |
. . . 4
#
# |
| 24 | 6, 15, 14 | mulassd 7971 |
. . . 4
#
# |
| 25 | divcanap1 8627 |
. . . . . . 7
# | |
| 26 | 25 | 3expb 1204 |
. . . . . 6
# |
| 27 | 26 | ad2ant2l 508 |
. . . . 5
#
# |
| 28 | 27 | oveq1d 5884 |
. . . 4
#
# |
| 29 | 23, 24, 28 | 3eqtr2d 2216 |
. . 3
#
# |
| 30 | 21, 29 | eqeq12d 2192 |
. 2
#
#
|
| 31 | 13, 30 | bitr3d 190 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1353 wcel 2148 class class class wbr 4000
(class class class)co 5869 cc 7800 cc0 7802 cmul 7807 # cap 8528
cdiv 8618 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 |
| This theorem is referenced by: divmuleqapd 8779 qtri3or 10229 |
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