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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 545 |
. . . . . . 7
# #
# | |
| 2 | simprll 543 |
. . . . . . 7
# #
# | |
| 3 | simprlr 544 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8998 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1278 |
. . . . . 6
# #
# |
| 6 | simpll 531 |
. . . . . . 7
# #
# | |
| 7 | simplrl 541 |
. . . . . . 7
# #
# | |
| 8 | simplrr 542 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8998 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1278 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8337 |
. . . . 5
# #
# |
| 12 | simplr 533 |
. . . . . 6
# #
# # | |
| 13 | simprl 535 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 9032 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1279 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2271 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 6091 |
. . 3
# #
# |
| 18 | simprr 537 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 9032 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1279 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8337 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 6090 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8336 |
. . . . . . . 8
# #
# |
| 24 | simprrr 546 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8976 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 9021 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 415 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2271 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2271 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 6090 |
. . . 4
# #
# |
| 31 | divclap 8998 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1278 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8339 |
. . . 4
# #
# |
| 34 | 10 | mullidd 8334 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2279 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2273 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8336 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8336 |
. . . 4
# #
# |
| 39 | mulap0 8972 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 514 |
. . . 4
# #
# #
|
| 41 | divclap 8998 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1278 |
. . 3
# #
# |
| 43 | divap0 9004 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8995 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1282 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1402 wcel 2209 class class class wbr 4125
(class class class)co 6075 cc 8167 cc0 8169 c1 8170 cmul 8174 # cap 8899
cdiv 8992 |
| 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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 |
| This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 |
| This theorem is referenced by: recdivap 9038 divcanap7 9041 divdivap1 9043 divdivap2 9044 divdivdivapi 9095 qreccl 10021 pcadd 13097 |
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