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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 8836 |
. . . . 5
# | |
| 2 | 1 | 3expb 1228 |
. . . 4
# |
| 3 | 2 | ad2ant2r 509 |
. . 3
#
# |
| 4 | divclap 8836 |
. . . . 5
# | |
| 5 | 4 | 3expb 1228 |
. . . 4
# |
| 6 | 5 | ad2ant2l 508 |
. . 3
#
# |
| 7 | mulcl 8137 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 509 |
. . . . 5
#
# |
| 9 | mulap0 8812 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | mulcanap2 8824 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1271 |
. 2
#
#
|
| 14 | simprll 537 |
. . . . 5
#
# | |
| 15 | simprrl 539 |
. . . . 5
#
# | |
| 16 | 3, 14, 15 | mulassd 8181 |
. . . 4
#
# |
| 17 | divcanap1 8839 |
. . . . . . 7
# | |
| 18 | 17 | 3expb 1228 |
. . . . . 6
# |
| 19 | 18 | ad2ant2r 509 |
. . . . 5
#
# |
| 20 | 19 | oveq1d 6022 |
. . . 4
#
# |
| 21 | 16, 20 | eqtr3d 2264 |
. . 3
#
# |
| 22 | 14, 15 | mulcomd 8179 |
. . . . 5
#
# |
| 23 | 22 | oveq2d 6023 |
. . . 4
#
# |
| 24 | 6, 15, 14 | mulassd 8181 |
. . . 4
#
# |
| 25 | divcanap1 8839 |
. . . . . . 7
# | |
| 26 | 25 | 3expb 1228 |
. . . . . 6
# |
| 27 | 26 | ad2ant2l 508 |
. . . . 5
#
# |
| 28 | 27 | oveq1d 6022 |
. . . 4
#
# |
| 29 | 23, 24, 28 | 3eqtr2d 2268 |
. . 3
#
# |
| 30 | 21, 29 | eqeq12d 2244 |
. 2
#
#
|
| 31 | 13, 30 | bitr3d 190 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1395 wcel 2200 class class class wbr 4083
(class class class)co 6007 cc 8008 cc0 8010 cmul 8015 # cap 8739
cdiv 8830 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 |
| This theorem is referenced by: divmuleqapd 8991 qtri3or 10472 |
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