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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 8637 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1238 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8637 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1238 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7981 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8671 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1239 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2210 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5893 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8671 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1239 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7981 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5892 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7980 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8617 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8660 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2210 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2210 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5892 |
. . . 4
# #
# |
| 31 | divclap 8637 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1238 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7983 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7978 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2218 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2212 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7980 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7980 |
. . . 4
# #
# |
| 39 | mulap0 8613 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8637 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1238 |
. . 3
# #
# |
| 43 | divap0 8643 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8634 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1242 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1353 wcel 2148 class class class wbr 4005
(class class class)co 5877 cc 7811 cc0 7813 c1 7814 cmul 7818 # cap 8540
cdiv 8631 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 |
| This theorem is referenced by: recdivap 8677 divcanap7 8680 divdivap1 8682 divdivap2 8683 divdivdivapi 8734 qreccl 9644 pcadd 12341 |
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