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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 9008 |
. . . . 5
# | |
| 2 | 1 | 3expb 1235 |
. . . 4
# |
| 3 | 2 | ad2ant2r 513 |
. . 3
#
# |
| 4 | divclap 9008 |
. . . . 5
# | |
| 5 | 4 | 3expb 1235 |
. . . 4
# |
| 6 | 5 | ad2ant2l 512 |
. . 3
#
# |
| 7 | mulcl 8306 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 513 |
. . . . 5
#
# |
| 9 | mulap0 8982 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | mulcanap2 8994 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1278 |
. 2
#
#
|
| 14 | simprll 543 |
. . . . 5
#
# | |
| 15 | simprrl 545 |
. . . . 5
#
# | |
| 16 | 3, 14, 15 | mulassd 8349 |
. . . 4
#
# |
| 17 | divcanap1 9011 |
. . . . . . 7
# | |
| 18 | 17 | 3expb 1235 |
. . . . . 6
# |
| 19 | 18 | ad2ant2r 513 |
. . . . 5
#
# |
| 20 | 19 | oveq1d 6100 |
. . . 4
#
# |
| 21 | 16, 20 | eqtr3d 2273 |
. . 3
#
# |
| 22 | 14, 15 | mulcomd 8347 |
. . . . 5
#
# |
| 23 | 22 | oveq2d 6101 |
. . . 4
#
# |
| 24 | 6, 15, 14 | mulassd 8349 |
. . . 4
#
# |
| 25 | divcanap1 9011 |
. . . . . . 7
# | |
| 26 | 25 | 3expb 1235 |
. . . . . 6
# |
| 27 | 26 | ad2ant2l 512 |
. . . . 5
#
# |
| 28 | 27 | oveq1d 6100 |
. . . 4
#
# |
| 29 | 23, 24, 28 | 3eqtr2d 2277 |
. . 3
#
# |
| 30 | 21, 29 | eqeq12d 2253 |
. 2
#
#
|
| 31 | 13, 30 | bitr3d 190 |
1
#
# |
| Colors of variables: wff set class |
| This proof depends on syntax axioms:
wi 4
wa 104
wb 105 wceq 1402 wcel 2209 class class class wbr 4130
(class class class)co 6085 cc 8177 cc0 8179 cmul 8184 # cap 8909
cdiv 9002 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 |
| This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-po 4441 df-iso 4442 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 |
| This theorem is used by: divmuleqapd 9163 qtri3or 10675 |
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