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| Mirrors > Home > ILE Home > Th. List > divmuldivap | Unicode version | ||
| Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| Ref | Expression |
|---|---|
| divmuldivap |
     ![/\](wedge.gif "/\"){kind=small}    #  
#     |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3anass 928 |
. . 3
# 
# | |
| 2 | 3anass 928 |
. . 3
#
# | |
| 3 | divclap 8145 |
. . . . . 6
# | |
| 4 | divclap 8145 |
. . . . . 6
# | |
| 5 | mulcl 7469 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 283 |
. . . . 5
#
# |
| 7 | mulcl 7469 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 493 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1102 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1099 |
. . . . 5
#
#
|
| 11 | mulap0 8123 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1102 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1099 |
. . . . 5
#
#
# |
| 14 | divcanap3 8165 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1174 |
. . . 4
#
# |
| 16 | simp2 944 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 300 |
. . . . . . 7
#
|
| 18 | simp2 944 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 300 |
. . . . . . 7
#
|
| 20 | mul4 7614 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 283 |
. . . . . 6
#
#
|
| 22 | divcanap2 8147 |
. . . . . . 7
#
| |
| 23 | divcanap2 8147 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5671 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2122 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5667 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2122 |
. . 3
#
# |
| 28 | 1, 2, 27 | syl2anbr 286 |
. 2
# # |
| 29 | 28 | an4s 555 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
w3a 924
wceq 1289
wcel 1438
class class class wbr 3845 (class class class)co 5652
cc 7348
cc0 7350
cmul 7355
# cap 8058 cdiv 8139 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulrcl 7444 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-precex 7455 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-apti 7460 ax-pre-ltadd 7461 ax-pre-mulgt0 7462 ax-pre-mulext 7463 |
| This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-reap 8052 df-ap 8059 df-div 8140 |
| This theorem is referenced by: divdivdivap 8180 divcanap5 8181 divmul13ap 8182 divmul24ap 8183 divmuldivapi 8239 divmuldivapd 8299 qmulcl 9122 mulexpzap 9995 expaddzap 9999 sqdivap 10019 |
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