Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > divadddivap | Unicode version |
Description: Addition of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) |
Ref | Expression |
---|---|
divadddivap | # # |
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 |
Copyright terms: Public domain | W3C validator |