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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 8147 |
. . . . 5
| |
| 2 | 1 | ad2ant2r 509 |
. . . 4
#
|
| 3 | 2 | adantrl 478 |
. . 3
#
# |
| 4 | mulcl 8147 |
. . . . 5
| |
| 5 | 4 | adantrr 479 |
. . . 4
# |
| 6 | 5 | ad2ant2lr 510 |
. . 3
#
# |
| 7 | mulcl 8147 |
. . . . . 6
| |
| 8 | 7 | ad2ant2r 509 |
. . . . 5
#
# |
| 9 | mulap0 8822 |
. . . . 5
#
# #
| |
| 10 | 8, 9 | jca 306 |
. . . 4
#
# # |
| 11 | 10 | adantl 277 |
. . 3
#
# # |
| 12 | divdirap 8865 |
. . 3
#
| |
| 13 | 3, 6, 11, 12 | syl3anc 1271 |
. 2
#
#
|
| 14 | simpll 527 |
. . . . . 6
#
# | |
| 15 | simprr 531 |
. . . . . . 7
#
# # | |
| 16 | 15 | simpld 112 |
. . . . . 6
#
# |
| 17 | 14, 16 | mulcomd 8189 |
. . . . 5
#
# |
| 18 | simprll 537 |
. . . . . 6
#
# | |
| 19 | 18, 16 | mulcomd 8189 |
. . . . 5
#
# |
| 20 | 17, 19 | oveq12d 6029 |
. . . 4
#
# |
| 21 | simprl 529 |
. . . . 5
#
# # | |
| 22 | divcanap5 8882 |
. . . . 5
#
# | |
| 23 | 14, 21, 15, 22 | syl3anc 1271 |
. . . 4
#
# |
| 24 | 20, 23 | eqtrd 2262 |
. . 3
#
# |
| 25 | simplr 528 |
. . . . . 6
#
# | |
| 26 | 25, 18 | mulcomd 8189 |
. . . . 5
#
# |
| 27 | 26 | oveq1d 6026 |
. . . 4
#
# |
| 28 | divcanap5 8882 |
. . . . 5
#
# | |
| 29 | 25, 15, 21, 28 | syl3anc 1271 |
. . . 4
#
# |
| 30 | 27, 29 | eqtrd 2262 |
. . 3
#
# |
| 31 | 24, 30 | oveq12d 6029 |
. 2
#
#
|
| 32 | 13, 31 | eqtr2d 2263 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1395
wcel 2200
class class class wbr 4084 (class class class)co 6011
cc 8018
cc0 8020
caddc 8023 cmul 8025 # cap 8749
cdiv 8840 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4203 ax-pow 4260 ax-pr 4295 ax-un 4526 ax-setind 4631 ax-cnex 8111 ax-resscn 8112 ax-1cn 8113 ax-1re 8114 ax-icn 8115 ax-addcl 8116 ax-addrcl 8117 ax-mulcl 8118 ax-mulrcl 8119 ax-addcom 8120 ax-mulcom 8121 ax-addass 8122 ax-mulass 8123 ax-distr 8124 ax-i2m1 8125 ax-0lt1 8126 ax-1rid 8127 ax-0id 8128 ax-rnegex 8129 ax-precex 8130 ax-cnre 8131 ax-pre-ltirr 8132 ax-pre-ltwlin 8133 ax-pre-lttrn 8134 ax-pre-apti 8135 ax-pre-ltadd 8136 ax-pre-mulgt0 8137 ax-pre-mulext 8138 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3890 df-br 4085 df-opab 4147 df-id 4386 df-po 4389 df-iso 4390 df-xp 4727 df-rel 4728 df-cnv 4729 df-co 4730 df-dm 4731 df-iota 5282 df-fun 5324 df-fv 5330 df-riota 5964 df-ov 6014 df-oprab 6015 df-mpo 6016 df-pnf 8204 df-mnf 8205 df-xr 8206 df-ltxr 8207 df-le 8208 df-sub 8340 df-neg 8341 df-reap 8743 df-ap 8750 df-div 8841 |
| This theorem is referenced by: divsubdivap 8896 divadddivapi 8942 qaddcl 9857 pcaddlem 12899 |
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