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| 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 8067 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1022 |
. 2
|
| 3 | cnre 8067 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1021 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8067 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1020 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simplrl 535 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 8100 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 529 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 8100 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 8092 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 536 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 8100 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 531 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 8100 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 8369 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 529 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 8100 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 8092 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 531 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 8100 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 8369 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8682 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1250 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8671 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1249 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8683 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8671 |
. . . . . . . . . . . . . . . . 17
# # | |
| 41 | 17, 30, 21, 40 | syl3anc 1249 |
. . . . . . . . . . . . . . . 16
# # |
| 42 | 39, 41 | sylbird 170 |
. . . . . . . . . . . . . . 15
# # |
| 43 | 37, 42 | orim12d 787 |
. . . . . . . . . . . . . 14
# # # # |
| 44 | 35, 43 | syld 45 |
. . . . . . . . . . . . 13
# # # |
| 45 | 15, 18 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 8092 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8682 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1250 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8672 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1249 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8672 |
. . . . . . . . . . . . . . . . 17
# # | |
| 54 | 10, 26, 21, 53 | syl3anc 1249 |
. . . . . . . . . . . . . . . 16
# # |
| 55 | 52, 54 | orim12d 787 |
. . . . . . . . . . . . . . 15
# # # # |
| 56 | 50, 55 | syld 45 |
. . . . . . . . . . . . . 14
# # # |
| 57 | orcom 729 |
. . . . . . . . . . . . . 14
# #
#
# | |
| 58 | 56, 57 | imbitrdi 161 |
. . . . . . . . . . . . 13
# # # |
| 59 | 44, 58 | jaod 718 |
. . . . . . . . . . . 12
# #
#
# |
| 60 | simpr 110 |
. . . . . . . . . . . . . . 15
| |
| 61 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 62 | 61 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 63 | 60, 62 | oveq12d 5961 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 534 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5961 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 4056 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8676 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1250 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8676 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1250 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 4056 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 8102 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8451 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 8101 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 8101 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 8102 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8451 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 8101 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 8102 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 8101 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8675 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1250 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 214 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8675 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1250 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 203 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 4056 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 169 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 115 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2630 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 115 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2630 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 115 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2630 |
. 2
# # |
| 102 | 2, 101 | mpd 13 |
1
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 709 w3a 980 wceq 1372 wcel 2175 wrex 2484 class class class wbr 4043
(class class class)co 5943 cc 7922 cr 7923 ci 7926 caddc 7927 cmul 7929 cneg 8243 # cap 8653 |
| 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 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-pre-mulext 8042 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-pnf 8108 df-mnf 8109 df-ltxr 8111 df-sub 8244 df-neg 8245 df-reap 8647 df-ap 8654 |
| This theorem is referenced by: mulext2 8685 mulext 8686 mulap0 8726 apmul1 8860 |
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