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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 8821 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1271 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8821 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1271 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8164 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8855 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1272 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2262 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 6016 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8855 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1272 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8164 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 6015 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8163 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8801 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8844 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2262 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2262 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 6015 |
. . . 4
# #
# |
| 31 | divclap 8821 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1271 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8166 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 8161 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2270 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2264 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8163 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8163 |
. . . 4
# #
# |
| 39 | mulap0 8797 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8821 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1271 |
. . 3
# #
# |
| 43 | divap0 8827 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8818 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1275 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1395 wcel 2200 class class class wbr 4082
(class class class)co 6000 cc 7993 cc0 7995 c1 7996 cmul 8000 # cap 8724
cdiv 8815 |
| 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 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113 |
| 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 3888 df-br 4083 df-opab 4145 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-iota 5277 df-fun 5319 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 |
| This theorem is referenced by: recdivap 8861 divcanap7 8864 divdivap1 8866 divdivap2 8867 divdivdivapi 8918 qreccl 9833 pcadd 12858 |
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