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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 977 |
. . 3
# 
# | |
| 2 | 3anass 977 |
. . 3
#
# | |
| 3 | divclap 8584 |
. . . . . 6
# | |
| 4 | divclap 8584 |
. . . . . 6
# | |
| 5 | mulcl 7890 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 287 |
. . . . 5
#
# |
| 7 | mulcl 7890 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 506 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1151 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1148 |
. . . . 5
#
#
|
| 11 | mulap0 8561 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1151 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1148 |
. . . . 5
#
#
# |
| 14 | divcanap3 8604 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1233 |
. . . 4
#
# |
| 16 | simp2 993 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 304 |
. . . . . . 7
#
|
| 18 | simp2 993 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 304 |
. . . . . . 7
#
|
| 20 | mul4 8040 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 287 |
. . . . . 6
#
#
|
| 22 | divcanap2 8586 |
. . . . . . 7
#
| |
| 23 | divcanap2 8586 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5870 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2205 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5866 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2205 |
. . 3
#
# |
| 28 | 1, 2, 27 | syl2anbr 290 |
. 2
# # |
| 29 | 28 | an4s 583 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
w3a 973
wceq 1348
wcel 2141
class class class wbr 3987 (class class class)co 5851
cc 7761
cc0 7763
cmul 7768
# cap 8489 cdiv 8578 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 |
| This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 |
| This theorem is referenced by: divdivdivap 8619 divcanap5 8620 divmul13ap 8621 divmul24ap 8622 divmuldivapi 8678 divmuldivapd 8738 qmulcl 9585 mulexpzap 10505 expaddzap 10509 sqdivap 10529 |
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