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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 8660 |
. . . . . 6
# | |
| 4 | divclap 8660 |
. . . . . 6
# | |
| 5 | mulcl 7963 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 289 |
. . . . 5
#
# |
| 7 | mulcl 7963 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 509 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1158 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1155 |
. . . . 5
#
#
|
| 11 | mulap0 8636 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1158 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1155 |
. . . . 5
#
#
# |
| 14 | divcanap3 8680 |
. . . . 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 8114 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 289 |
. . . . . 6
#
#
|
| 22 | divcanap2 8662 |
. . . . . . 7
#
| |
| 23 | divcanap2 8662 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5911 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2224 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5907 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2224 |
. . 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 2160
class class class wbr 4018 (class class class)co 5892
cc 7834
cc0 7836
cmul 7841
# cap 8563 cdiv 8654 |
| 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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 |
| 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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5234 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 |
| This theorem is referenced by: divdivdivap 8695 divcanap5 8696 divmul13ap 8697 divmul24ap 8698 divmuldivapi 8754 divmuldivapd 8814 qmulcl 9662 mulexpzap 10586 expaddzap 10590 sqdivap 10610 |
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