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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 541 |
. . . . . . 7
# #
# | |
| 2 | simprll 539 |
. . . . . . 7
# #
# | |
| 3 | simprlr 540 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8969 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1274 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 537 |
. . . . . . 7
# #
# | |
| 8 | simplrr 538 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8969 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1274 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8311 |
. . . . 5
# #
# |
| 12 | simplr 529 |
. . . . . 6
# #
# # | |
| 13 | simprl 531 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 9003 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1275 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2267 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 6074 |
. . 3
# #
# |
| 18 | simprr 533 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 9003 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1275 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8311 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 6073 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8310 |
. . . . . . . 8
# #
# |
| 24 | simprrr 542 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8949 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8992 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2267 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2267 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 6073 |
. . . 4
# #
# |
| 31 | divclap 8969 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1274 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8313 |
. . . 4
# #
# |
| 34 | 10 | mullidd 8308 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2275 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2269 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8310 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8310 |
. . . 4
# #
# |
| 39 | mulap0 8945 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8969 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1274 |
. . 3
# #
# |
| 43 | divap0 8975 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8966 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1278 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1398 wcel 2205 class class class wbr 4114
(class class class)co 6058 cc 8141 cc0 8143 c1 8144 cmul 8148 # cap 8872
cdiv 8963 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 |
| This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 |
| This theorem is referenced by: recdivap 9009 divcanap7 9012 divdivap1 9014 divdivap2 9015 divdivdivapi 9066 qreccl 9992 pcadd 13063 |
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