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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 7913 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 509 |
. . . 4
#
|
| 3 | 2 | adantrl 478 |
. . 3
#
# |
| 4 | mulcl 7913 |
. . . . 5
| |
| 5 | 4 | adantrr 479 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 510 |
. . 3
#
# |
| 7 | mulcl 7913 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 509 |
. . . . 5
#
# |
| 9 | mulap0 8584 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | divdirap 8627 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1238 |
. 2
#
#
|
| 14 | simpll 527 |
. . . . . 6
#
# | |
| 15 | simprr 531 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 112 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 7953 |
. . . . 5
#
# |
| 18 | simprll 537 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 7953 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 5883 |
. . . 4
#
# |
| 21 | simprl 529 |
. . . . 5
#
# # | |
| 22 | divcanap5 8644 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1238 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2208 |
. . 3
#
# |
| 25 | simplr 528 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 7953 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 5880 |
. . . 4
#
# |
| 28 | divcanap5 8644 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1238 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2208 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 5883 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2209 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1353
wcel 2146
class class class wbr 3998 (class class class)co 5865
cc 7784
cc0 7786
caddc 7789 cmul 7791 # cap 8512
cdiv 8602 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 |
| This theorem is referenced by: divsubdivap 8658 divadddivapi 8704 qaddcl 9608 pcaddlem 12305 |
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