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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 7686 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 987 |
. 2
|
| 3 | cnre 7686 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 986 |
. . . . . 6
|
| 5 | 4 | ad2antrr 477 |
. . . . 5
|
| 6 | cnre 7686 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 985 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 272 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 481 |
. . . . . . . 8
|
| 10 | simplrl 507 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 7718 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 503 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 477 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 481 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 7718 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 7710 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 508 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 7718 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 504 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 477 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 481 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 7718 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 7983 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 503 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 481 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 7718 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 7710 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 504 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 481 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 7718 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 7983 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8290 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1200 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8279 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1199 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8291 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 406 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8279 |
. . . . . . . . . . . . . . . . 17
# # | |
| 41 | 17, 30, 21, 40 | syl3anc 1199 |
. . . . . . . . . . . . . . . 16
# # |
| 42 | 39, 41 | sylbird 169 |
. . . . . . . . . . . . . . 15
# # |
| 43 | 37, 42 | orim12d 758 |
. . . . . . . . . . . . . 14
# # # # |
| 44 | 35, 43 | syld 45 |
. . . . . . . . . . . . 13
# # # |
| 45 | 15, 18 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 7710 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8290 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1200 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8280 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1199 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8280 |
. . . . . . . . . . . . . . . . 17
# # | |
| 54 | 10, 26, 21, 53 | syl3anc 1199 |
. . . . . . . . . . . . . . . 16
# # |
| 55 | 52, 54 | orim12d 758 |
. . . . . . . . . . . . . . 15
# # # # |
| 56 | 50, 55 | syld 45 |
. . . . . . . . . . . . . 14
# # # |
| 57 | orcom 700 |
. . . . . . . . . . . . . 14
# #
#
# | |
| 58 | 56, 57 | syl6ib 160 |
. . . . . . . . . . . . 13
# # # |
| 59 | 44, 58 | jaod 689 |
. . . . . . . . . . . 12
# #
#
# |
| 60 | simpr 109 |
. . . . . . . . . . . . . . 15
| |
| 61 | simplr 502 |
. . . . . . . . . . . . . . . 16
| |
| 62 | 61 | ad3antrrr 481 |
. . . . . . . . . . . . . . 15
|
| 63 | 60, 62 | oveq12d 5746 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 506 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5746 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 3908 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8284 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1200 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8284 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1200 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 3908 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 7720 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8061 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 7719 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 7719 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 7720 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8061 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 7719 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 7720 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 7719 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8283 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1200 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 213 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8283 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1200 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 202 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 3908 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 168 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 114 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2531 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 114 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2531 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 114 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2531 |
. 2
# # |
| 102 | 2, 101 | mpd 13 |
1
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 680 w3a 945 wceq 1314 wcel 1463 wrex 2391 class class class wbr 3895
(class class class)co 5728 cc 7545 cr 7546 ci 7549 caddc 7550 cmul 7552 cneg 7857 # cap 8261 |
| 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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 ax-pre-mulext 7663 |
| This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-ltxr 7729 df-sub 7858 df-neg 7859 df-reap 8255 df-ap 8262 |
| This theorem is referenced by: mulext2 8293 mulext 8294 mulap0 8328 apmul1 8461 |
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