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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 7766 |
. . . . 5
# | |
| 2 | 1 | 3expb 1139 |
. . . 4
# |
| 3 | 2 | ad2ant2r 492 |
. . 3
#
# |
| 4 | divclap 7766 |
. . . . 5
# | |
| 5 | 4 | 3expb 1139 |
. . . 4
# |
| 6 | 5 | ad2ant2l 491 |
. . 3
#
# |
| 7 | mulcl 7100 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 492 |
. . . . 5
#
# |
| 9 | mulap0 7744 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 300 |
. . . 4
#
# # |
| 11 | 10 | adantl 271 |
. . 3
#
# # |
| 12 | mulcanap2 7756 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1169 |
. 2
#
#
|
| 14 | simprll 503 |
. . . . 5
#
# | |
| 15 | simprrl 505 |
. . . . 5
#
# | |
| 16 | 3, 14, 15 | mulassd 7142 |
. . . 4
#
# |
| 17 | divcanap1 7769 |
. . . . . . 7
# | |
| 18 | 17 | 3expb 1139 |
. . . . . 6
# |
| 19 | 18 | ad2ant2r 492 |
. . . . 5
#
# |
| 20 | 19 | oveq1d 5547 |
. . . 4
#
# |
| 21 | 16, 20 | eqtr3d 2115 |
. . 3
#
# |
| 22 | 14, 15 | mulcomd 7140 |
. . . . 5
#
# |
| 23 | 22 | oveq2d 5548 |
. . . 4
#
# |
| 24 | 6, 15, 14 | mulassd 7142 |
. . . 4
#
# |
| 25 | divcanap1 7769 |
. . . . . . 7
# | |
| 26 | 25 | 3expb 1139 |
. . . . . 6
# |
| 27 | 26 | ad2ant2l 491 |
. . . . 5
#
# |
| 28 | 27 | oveq1d 5547 |
. . . 4
#
# |
| 29 | 23, 24, 28 | 3eqtr2d 2119 |
. . 3
#
# |
| 30 | 21, 29 | eqeq12d 2095 |
. 2
#
#
|
| 31 | 13, 30 | bitr3d 188 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wb 103 wceq 1284 wcel 1433 class class class wbr 3785
(class class class)co 5532 cc 6979 cc0 6981 cmul 6986 # cap 7681
cdiv 7760 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
| This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 |
| This theorem is referenced by: qtri3or 9252 |
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