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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 8952 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1274 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 537 |
. . . . . . 7
# #
# | |
| 8 | simplrr 538 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8952 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1274 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8295 |
. . . . 5
# #
# |
| 12 | simplr 529 |
. . . . . 6
# #
# # | |
| 13 | simprl 531 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8986 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1275 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2265 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 6066 |
. . 3
# #
# |
| 18 | simprr 533 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8986 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1275 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8295 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 6065 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8294 |
. . . . . . . 8
# #
# |
| 24 | simprrr 542 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8932 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8975 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2265 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2265 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 6065 |
. . . 4
# #
# |
| 31 | divclap 8952 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1274 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8297 |
. . . 4
# #
# |
| 34 | 10 | mullidd 8292 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2273 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2267 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8294 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8294 |
. . . 4
# #
# |
| 39 | mulap0 8928 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8952 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1274 |
. . 3
# #
# |
| 43 | divap0 8958 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8949 |
. . 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 2203 class class class wbr 4109
(class class class)co 6050 cc 8125 cc0 8127 c1 8128 cmul 8132 # cap 8855
cdiv 8946 |
| 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 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 |
| 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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-po 4417 df-iso 4418 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 |
| This theorem is referenced by: recdivap 8992 divcanap7 8995 divdivap1 8997 divdivap2 8998 divdivdivapi 9049 qreccl 9974 pcadd 13038 |
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