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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 7619 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 496 |
. . . 4
#
|
| 3 | 2 | adantrl 465 |
. . 3
#
# |
| 4 | mulcl 7619 |
. . . . 5
| |
| 5 | 4 | adantrr 466 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 497 |
. . 3
#
# |
| 7 | mulcl 7619 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 496 |
. . . . 5
#
# |
| 9 | mulap0 8276 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 302 |
. . . 4
#
# # |
| 11 | 10 | adantl 273 |
. . 3
#
# # |
| 12 | divdirap 8318 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1184 |
. 2
#
#
|
| 14 | simpll 499 |
. . . . . 6
#
# | |
| 15 | simprr 502 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 111 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 7659 |
. . . . 5
#
# |
| 18 | simprll 507 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 7659 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 5724 |
. . . 4
#
# |
| 21 | simprl 501 |
. . . . 5
#
# # | |
| 22 | divcanap5 8335 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1184 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2132 |
. . 3
#
# |
| 25 | simplr 500 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 7659 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 5721 |
. . . 4
#
# |
| 28 | divcanap5 8335 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1184 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2132 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 5724 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2133 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1299
wcel 1448
class class class wbr 3875 (class class class)co 5706
cc 7498
cc0 7500
caddc 7503 cmul 7505 # cap 8209
cdiv 8293 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 |
| This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 |
| This theorem is referenced by: divsubdivap 8349 divadddivapi 8395 qaddcl 9277 |
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