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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 984 |
. . 3
# 
# | |
| 2 | 3anass 984 |
. . 3
#
# | |
| 3 | divclap 8697 |
. . . . . 6
# | |
| 4 | divclap 8697 |
. . . . . 6
# | |
| 5 | mulcl 7999 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 289 |
. . . . 5
#
# |
| 7 | mulcl 7999 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 509 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1158 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1155 |
. . . . 5
#
#
|
| 11 | mulap0 8673 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1158 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1155 |
. . . . 5
#
#
# |
| 14 | divcanap3 8717 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1249 |
. . . 4
#
# |
| 16 | simp2 1000 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 306 |
. . . . . . 7
#
|
| 18 | simp2 1000 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 306 |
. . . . . . 7
#
|
| 20 | mul4 8151 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 289 |
. . . . . 6
#
#
|
| 22 | divcanap2 8699 |
. . . . . . 7
#
| |
| 23 | divcanap2 8699 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5937 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2228 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5933 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2228 |
. . 3
#
# |
| 28 | 1, 2, 27 | syl2anbr 292 |
. 2
# # |
| 29 | 28 | an4s 588 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 980
wceq 1364
wcel 2164
class class class wbr 4029 (class class class)co 5918
cc 7870
cc0 7872
cmul 7877
# cap 8600 cdiv 8691 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 |
| This theorem is referenced by: divdivdivap 8732 divcanap5 8733 divmul13ap 8734 divmul24ap 8735 divmuldivapi 8791 divmuldivapd 8851 qmulcl 9702 mulexpzap 10650 expaddzap 10654 sqdivap 10674 |
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