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| Mirrors > Home > ILE Home > Th. List > divadddivap | Unicode version | ||
| Description: Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| Ref | Expression |
|---|---|
| divadddivap |
     ![/\](wedge.gif "/\"){kind=small}    #  
#      |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulcl 7871 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 501 |
. . . 4
#
|
| 3 | 2 | adantrl 470 |
. . 3
#
# |
| 4 | mulcl 7871 |
. . . . 5
| |
| 5 | 4 | adantrr 471 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 502 |
. . 3
#
# |
| 7 | mulcl 7871 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 501 |
. . . . 5
#
# |
| 9 | mulap0 8542 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 304 |
. . . 4
#
# # |
| 11 | 10 | adantl 275 |
. . 3
#
# # |
| 12 | divdirap 8584 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1227 |
. 2
#
#
|
| 14 | simpll 519 |
. . . . . 6
#
# | |
| 15 | simprr 522 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 111 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 7911 |
. . . . 5
#
# |
| 18 | simprll 527 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 7911 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 5854 |
. . . 4
#
# |
| 21 | simprl 521 |
. . . . 5
#
# # | |
| 22 | divcanap5 8601 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1227 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2197 |
. . 3
#
# |
| 25 | simplr 520 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 7911 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 5851 |
. . . 4
#
# |
| 28 | divcanap5 8601 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1227 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2197 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 5854 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2198 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1342
wcel 2135
class class class wbr 3976 (class class class)co 5836
cc 7742
cc0 7744
caddc 7747 cmul 7749 # cap 8470
cdiv 8559 |
| 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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
| This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 |
| This theorem is referenced by: divsubdivap 8615 divadddivapi 8661 qaddcl 9564 pcaddlem 12247 |
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