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 7857 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1005 |
. 2
|
| 3 | cnre 7857 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1004 |
. . . . . 6
|
| 5 | 4 | ad2antrr 480 |
. . . . 5
|
| 6 | cnre 7857 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1003 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 274 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 484 |
. . . . . . . 8
|
| 10 | simplrl 525 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 7889 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 521 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 480 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 484 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 7889 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 7881 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 526 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 7889 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 522 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 480 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 484 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 7889 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 8156 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 521 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 484 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 7889 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 7881 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 522 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 484 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 7889 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 8156 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8468 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1221 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8457 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1220 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8469 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 409 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8457 |
. . . . . . . . . . . . . . . . 17
# # | |
| 41 | 17, 30, 21, 40 | syl3anc 1220 |
. . . . . . . . . . . . . . . 16
# # |
| 42 | 39, 41 | sylbird 169 |
. . . . . . . . . . . . . . 15
# # |
| 43 | 37, 42 | orim12d 776 |
. . . . . . . . . . . . . 14
# # # # |
| 44 | 35, 43 | syld 45 |
. . . . . . . . . . . . 13
# # # |
| 45 | 15, 18 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 7881 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8468 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1221 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8458 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1220 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8458 |
. . . . . . . . . . . . . . . . 17
# # | |
| 54 | 10, 26, 21, 53 | syl3anc 1220 |
. . . . . . . . . . . . . . . 16
# # |
| 55 | 52, 54 | orim12d 776 |
. . . . . . . . . . . . . . 15
# # # # |
| 56 | 50, 55 | syld 45 |
. . . . . . . . . . . . . 14
# # # |
| 57 | orcom 718 |
. . . . . . . . . . . . . 14
# #
#
# | |
| 58 | 56, 57 | syl6ib 160 |
. . . . . . . . . . . . 13
# # # |
| 59 | 44, 58 | jaod 707 |
. . . . . . . . . . . 12
# #
#
# |
| 60 | simpr 109 |
. . . . . . . . . . . . . . 15
| |
| 61 | simplr 520 |
. . . . . . . . . . . . . . . 16
| |
| 62 | 61 | ad3antrrr 484 |
. . . . . . . . . . . . . . 15
|
| 63 | 60, 62 | oveq12d 5836 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 524 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5836 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 3978 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8462 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1221 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8462 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1221 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 3978 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 7891 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8238 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 7890 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 7890 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 7891 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8238 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 7890 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 7891 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 7890 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8461 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1221 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 213 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8461 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1221 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 202 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 3978 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 168 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 114 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2582 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 114 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2582 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 114 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2582 |
. 2
# # |
| 102 | 2, 101 | mpd 13 |
1
# # |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 698 w3a 963 wceq 1335 wcel 2128 wrex 2436 class class class wbr 3965
(class class class)co 5818 cc 7713 cr 7714 ci 7717 caddc 7718 cmul 7720 cneg 8030 # cap 8439 |
| 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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-cnex 7806 ax-resscn 7807 ax-1cn 7808 ax-1re 7809 ax-icn 7810 ax-addcl 7811 ax-addrcl 7812 ax-mulcl 7813 ax-mulrcl 7814 ax-addcom 7815 ax-mulcom 7816 ax-addass 7817 ax-mulass 7818 ax-distr 7819 ax-i2m1 7820 ax-0lt1 7821 ax-1rid 7822 ax-0id 7823 ax-rnegex 7824 ax-precex 7825 ax-cnre 7826 ax-pre-ltirr 7827 ax-pre-ltwlin 7828 ax-pre-lttrn 7829 ax-pre-apti 7830 ax-pre-ltadd 7831 ax-pre-mulgt0 7832 ax-pre-mulext 7833 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4252 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-iota 5132 df-fun 5169 df-fv 5175 df-riota 5774 df-ov 5821 df-oprab 5822 df-mpo 5823 df-pnf 7897 df-mnf 7898 df-ltxr 7900 df-sub 8031 df-neg 8032 df-reap 8433 df-ap 8440 |
| This theorem is referenced by: mulext2 8471 mulext 8472 mulap0 8511 apmul1 8644 |
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