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| Mirrors > Home > ILE Home > Th. List > divdivdivap | Unicode version | ||
| Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| Ref | Expression |
|---|---|
| divdivdivap |
     ![/\](wedge.gif "/\"){kind=small}  #    # 
#     |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simprrl 539 |
. . . . . . 7
# #
# | |
| 2 | simprll 537 |
. . . . . . 7
# #
# | |
| 3 | simprlr 538 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8771 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1250 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8771 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1250 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8114 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8805 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1251 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2239 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5973 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8805 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1251 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8114 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5972 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8113 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8751 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8794 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2239 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2239 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5972 |
. . . 4
# #
# |
| 31 | divclap 8771 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1250 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8116 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 8111 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2247 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2241 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8113 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8113 |
. . . 4
# #
# |
| 39 | mulap0 8747 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8771 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1250 |
. . 3
# #
# |
| 43 | divap0 8777 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8768 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1254 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1373 wcel 2177 class class class wbr 4051
(class class class)co 5957 cc 7943 cc0 7945 c1 7946 cmul 7950 # cap 8674
cdiv 8765 |
| 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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-br 4052 df-opab 4114 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-iota 5241 df-fun 5282 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 |
| This theorem is referenced by: recdivap 8811 divcanap7 8814 divdivap1 8816 divdivap2 8817 divdivdivapi 8868 qreccl 9783 pcadd 12738 |
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