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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 505 |
. . . . . . 7
# #
# | |
| 2 | simprll 503 |
. . . . . . 7
# #
# | |
| 3 | simprlr 504 |
. . . . . . 7
# #
# # | |
| 4 | divclap 7766 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1169 |
. . . . . 6
# #
# |
| 6 | simpll 495 |
. . . . . . 7
# #
# | |
| 7 | simplrl 501 |
. . . . . . 7
# #
# | |
| 8 | simplrr 502 |
. . . . . . 7
# #
# # | |
| 9 | divclap 7766 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1169 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7140 |
. . . . 5
# #
# |
| 12 | simplr 496 |
. . . . . 6
# #
# # | |
| 13 | simprl 497 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 7800 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1170 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2113 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5548 |
. . 3
# #
# |
| 18 | simprr 498 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 7800 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1170 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7140 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5547 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7139 |
. . . . . . . 8
# #
# |
| 24 | simprrr 506 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 7748 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 7789 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 403 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2113 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2113 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5547 |
. . . 4
# #
# |
| 31 | divclap 7766 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1169 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7142 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7137 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2121 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2115 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7139 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7139 |
. . . 4
# #
# |
| 39 | mulap0 7744 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 493 |
. . . 4
# #
# #
|
| 41 | divclap 7766 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1169 |
. . 3
# #
# |
| 43 | divap0 7772 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 271 |
. . 3
# #
# #
|
| 45 | divmulap 7763 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1173 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 165 |
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 c1 6982 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: recdivap 7806 divcanap7 7809 divdivap1 7811 divdivap2 7812 divdivdivapi 7863 qreccl 8727 |
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