Nearby theorems
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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 7755 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1004 |
. 2
|
| 3 | cnre 7755 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1003 |
. . . . . 6
|
| 5 | 4 | ad2antrr 479 |
. . . . 5
|
| 6 | cnre 7755 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1002 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 274 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 483 |
. . . . . . . 8
|
| 10 | simplrl 524 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 7787 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 520 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 479 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 483 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 7787 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 7779 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 525 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 7787 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 521 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 479 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 483 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 7787 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 8053 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 520 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 483 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 7787 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 7779 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 521 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 483 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 7787 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 8053 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8365 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1217 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8354 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1216 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8366 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 408 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8354 |
. . . . . . . . . . . . . . . . 17
# # | |
| 41 | 17, 30, 21, 40 | syl3anc 1216 |
. . . . . . . . . . . . . . . 16
# # |
| 42 | 39, 41 | sylbird 169 |
. . . . . . . . . . . . . . 15
# # |
| 43 | 37, 42 | orim12d 775 |
. . . . . . . . . . . . . 14
# # # # |
| 44 | 35, 43 | syld 45 |
. . . . . . . . . . . . 13
# # # |
| 45 | 15, 18 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 7779 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8365 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1217 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8355 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1216 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8355 |
. . . . . . . . . . . . . . . . 17
# # | |
| 54 | 10, 26, 21, 53 | syl3anc 1216 |
. . . . . . . . . . . . . . . 16
# # |
| 55 | 52, 54 | orim12d 775 |
. . . . . . . . . . . . . . 15
# # # # |
| 56 | 50, 55 | syld 45 |
. . . . . . . . . . . . . 14
# # # |
| 57 | orcom 717 |
. . . . . . . . . . . . . 14
# #
#
# | |
| 58 | 56, 57 | syl6ib 160 |
. . . . . . . . . . . . 13
# # # |
| 59 | 44, 58 | jaod 706 |
. . . . . . . . . . . 12
# #
#
# |
| 60 | simpr 109 |
. . . . . . . . . . . . . . 15
| |
| 61 | simplr 519 |
. . . . . . . . . . . . . . . 16
| |
| 62 | 61 | ad3antrrr 483 |
. . . . . . . . . . . . . . 15
|
| 63 | 60, 62 | oveq12d 5785 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 523 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5785 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 3937 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8359 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1217 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8359 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1217 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 3937 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 7789 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8135 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 7788 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 7788 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 7789 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8135 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 7788 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 7789 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 7788 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8358 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1217 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 213 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8358 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1217 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 202 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 3937 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 168 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 114 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2555 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 114 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2555 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 114 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2555 |
. 2
# # |
| 102 | 2, 101 | mpd 13 |
1
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924
(class class class)co 5767 cc 7611 cr 7612 ci 7615 caddc 7616 cmul 7618 cneg 7927 # cap 8336 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
| This theorem is referenced by: mulext2 8368 mulext 8369 mulap0 8408 apmul1 8541 |
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