Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > mulext1 | Unicode version | ||
| Description: Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
| Ref | Expression |
|---|---|
| mulext1 |
     ![/\](wedge.gif "/\"){kind=small} 
    # # |
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnre 7950 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1020 |
. 2
|
| 3 | cnre 7950 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1019 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 7950 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1018 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simplrl 535 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 7982 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 529 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 7982 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 7974 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 536 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 7982 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 531 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 7982 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 8251 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 529 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 7982 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 7974 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 531 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 7982 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 8251 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8563 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1239 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8552 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1238 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8564 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8552 |
. . . . . . . . . . . . . . . . 17
# # | |
| 41 | 17, 30, 21, 40 | syl3anc 1238 |
. . . . . . . . . . . . . . . 16
# # |
| 42 | 39, 41 | sylbird 170 |
. . . . . . . . . . . . . . 15
# # |
| 43 | 37, 42 | orim12d 786 |
. . . . . . . . . . . . . 14
# # # # |
| 44 | 35, 43 | syld 45 |
. . . . . . . . . . . . 13
# # # |
| 45 | 15, 18 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 7974 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8563 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1239 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8553 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1238 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8553 |
. . . . . . . . . . . . . . . . 17
# # | |
| 54 | 10, 26, 21, 53 | syl3anc 1238 |
. . . . . . . . . . . . . . . 16
# # |
| 55 | 52, 54 | orim12d 786 |
. . . . . . . . . . . . . . 15
# # # # |
| 56 | 50, 55 | syld 45 |
. . . . . . . . . . . . . 14
# # # |
| 57 | orcom 728 |
. . . . . . . . . . . . . 14
# #
#
# | |
| 58 | 56, 57 | syl6ib 161 |
. . . . . . . . . . . . 13
# # # |
| 59 | 44, 58 | jaod 717 |
. . . . . . . . . . . 12
# #
#
# |
| 60 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 61 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 62 | 61 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 63 | 60, 62 | oveq12d 5890 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 534 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5890 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 4015 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8557 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1239 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8557 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1239 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 4015 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 7984 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8333 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 7983 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 7983 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 7984 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8333 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 7983 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 7984 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 7983 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8556 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1239 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 214 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8556 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1239 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 203 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 4015 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 169 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 115 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2602 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 115 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2602 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 115 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2602 |
. 2
# # |
| 102 | 2, 101 | mpd 13 |
1
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 708 w3a 978 wceq 1353 wcel 2148 wrex 2456 class class class wbr 4002
(class class class)co 5872 cc 7806 cr 7807 ci 7810 caddc 7811 cmul 7813 cneg 8125 # cap 8534 |
| 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 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 |
| This theorem is referenced by: mulext2 8566 mulext 8567 mulap0 8607 apmul1 8741 |
| Copyright terms: Public domain | W3C validator |