Nearby theorems
|
|
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 |
     ![/\](wedge.gif "/\"){kind=small}    #  
#      |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulcl 7162 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 493 |
. . . 4
#
|
| 3 | 2 | adantrl 462 |
. . 3
#
# |
| 4 | mulcl 7162 |
. . . . 5
| |
| 5 | 4 | adantrr 463 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 494 |
. . 3
#
# |
| 7 | mulcl 7162 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 493 |
. . . . 5
#
# |
| 9 | mulap0 7811 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 300 |
. . . 4
#
# # |
| 11 | 10 | adantl 271 |
. . 3
#
# # |
| 12 | divdirap 7852 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1170 |
. 2
#
#
|
| 14 | simpll 496 |
. . . . . 6
#
# | |
| 15 | simprr 499 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 110 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 7202 |
. . . . 5
#
# |
| 18 | simprll 504 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 7202 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 5561 |
. . . 4
#
# |
| 21 | simprl 498 |
. . . . 5
#
# # | |
| 22 | divcanap5 7869 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1170 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2114 |
. . 3
#
# |
| 25 | simplr 497 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 7202 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 5558 |
. . . 4
#
# |
| 28 | divcanap5 7869 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1170 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2114 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 5561 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2115 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wceq 1285
wcel 1434
class class class wbr 3793 (class class class)co 5543
cc 7041
cc0 7043
caddc 7046 cmul 7048 # cap 7748
cdiv 7827 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-mulrcl 7137 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-precex 7148 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 ax-pre-mulgt0 7155 ax-pre-mulext 7156 |
| This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rmo 2357 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-id 4056 df-po 4059 df-iso 4060 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-iota 4897 df-fun 4934 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-reap 7742 df-ap 7749 df-div 7828 |
| This theorem is referenced by: divsubdivap 7883 divadddivapi 7929 qaddcl 8801 |
| Copyright terms: Public domain | W3C validator |