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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 8052 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 509 |
. . . 4
#
|
| 3 | 2 | adantrl 478 |
. . 3
#
# |
| 4 | mulcl 8052 |
. . . . 5
| |
| 5 | 4 | adantrr 479 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 510 |
. . 3
#
# |
| 7 | mulcl 8052 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 509 |
. . . . 5
#
# |
| 9 | mulap0 8727 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | divdirap 8770 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1250 |
. 2
#
#
|
| 14 | simpll 527 |
. . . . . 6
#
# | |
| 15 | simprr 531 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 112 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 8094 |
. . . . 5
#
# |
| 18 | simprll 537 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 8094 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 5962 |
. . . 4
#
# |
| 21 | simprl 529 |
. . . . 5
#
# # | |
| 22 | divcanap5 8787 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1250 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2238 |
. . 3
#
# |
| 25 | simplr 528 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 8094 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 5959 |
. . . 4
#
# |
| 28 | divcanap5 8787 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1250 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2238 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 5962 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2239 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1373
wcel 2176
class class class wbr 4044 (class class class)co 5944
cc 7923
cc0 7925
caddc 7928 cmul 7930 # cap 8654
cdiv 8745 |
| 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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-po 4343 df-iso 4344 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 |
| This theorem is referenced by: divsubdivap 8801 divadddivapi 8847 qaddcl 9756 pcaddlem 12662 |
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