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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 7989 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1022 |
. 2
|
| 3 | cnre 7989 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1021 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 7989 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1020 |
. . . . . . . . . 10
|
| 8 | 7 | adantr 276 |
. . . . . . . . 9
|
| 9 | 8 | ad3antrrr 492 |
. . . . . . . 8
|
| 10 | simplrl 535 |
. . . . . . . . . . . . . . . . 17
| |
| 11 | 10 | recnd 8022 |
. . . . . . . . . . . . . . . 16
|
| 12 | simprl 529 |
. . . . . . . . . . . . . . . . . . 19
| |
| 13 | 12 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . 18
|
| 14 | 13 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 15 | 14 | recnd 8022 |
. . . . . . . . . . . . . . . 16
|
| 16 | 11, 15 | mulcld 8014 |
. . . . . . . . . . . . . . 15
|
| 17 | simplrr 536 |
. . . . . . . . . . . . . . . . . 18
| |
| 18 | 17 | recnd 8022 |
. . . . . . . . . . . . . . . . 17
|
| 19 | simprr 531 |
. . . . . . . . . . . . . . . . . . . 20
| |
| 20 | 19 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . 19
|
| 21 | 20 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 22 | 21 | recnd 8022 |
. . . . . . . . . . . . . . . . 17
|
| 23 | 18, 22 | mulcld 8014 |
. . . . . . . . . . . . . . . 16
|
| 24 | 23 | negcld 8291 |
. . . . . . . . . . . . . . 15
 |
| 25 | simprl 529 |
. . . . . . . . . . . . . . . . . 18
| |
| 26 | 25 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . 17
|
| 27 | 26 | recnd 8022 |
. . . . . . . . . . . . . . . 16
|
| 28 | 27, 15 | mulcld 8014 |
. . . . . . . . . . . . . . 15
|
| 29 | simprr 531 |
. . . . . . . . . . . . . . . . . . 19
| |
| 30 | 29 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
|
| 31 | 30 | recnd 8022 |
. . . . . . . . . . . . . . . . 17
|
| 32 | 31, 22 | mulcld 8014 |
. . . . . . . . . . . . . . . 16
|
| 33 | 32 | negcld 8291 |
. . . . . . . . . . . . . . 15
|
| 34 | addext 8603 |
. . . . . . . . . . . . . . 15
#
#  # | |
| 35 | 16, 24, 28, 33, 34 | syl22anc 1250 |
. . . . . . . . . . . . . 14
# # # |
| 36 | remulext1 8592 |
. . . . . . . . . . . . . . . 16
# # | |
| 37 | 10, 26, 14, 36 | syl3anc 1249 |
. . . . . . . . . . . . . . 15
# # |
| 38 | apneg 8604 |
. . . . . . . . . . . . . . . . 17
#  # | |
| 39 | 23, 32, 38 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
#
#
|
| 40 | remulext1 8592 |
. . . . . . . . . . . . . . . . 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 8014 |
. . . . . . . . . . . . . . . 16
|
| 46 | 22, 11 | mulcld 8014 |
. . . . . . . . . . . . . . . 16
|
| 47 | 15, 31 | mulcld 8014 |
. . . . . . . . . . . . . . . 16
|
| 48 | 22, 27 | mulcld 8014 |
. . . . . . . . . . . . . . . 16
|
| 49 | addext 8603 |
. . . . . . . . . . . . . . . 16
# # # | |
| 50 | 45, 46, 47, 48, 49 | syl22anc 1250 |
. . . . . . . . . . . . . . 15
# # # |
| 51 | remulext2 8593 |
. . . . . . . . . . . . . . . . 17
# # | |
| 52 | 17, 30, 14, 51 | syl3anc 1249 |
. . . . . . . . . . . . . . . 16
# # |
| 53 | remulext2 8593 |
. . . . . . . . . . . . . . . . 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 5918 |
. . . . . . . . . . . . . 14
|
| 64 | simpllr 534 |
. . . . . . . . . . . . . . 15
| |
| 65 | 64, 62 | oveq12d 5918 |
. . . . . . . . . . . . . 14
|
| 66 | 63, 65 | breq12d 4034 |
. . . . . . . . . . . . 13
#
# |
| 67 | mulreim 8597 |
. . . . . . . . . . . . . . 15
| |
| 68 | 10, 17, 14, 21, 67 | syl22anc 1250 |
. . . . . . . . . . . . . 14
|
| 69 | mulreim 8597 |
. . . . . . . . . . . . . . 15
| |
| 70 | 26, 30, 14, 21, 69 | syl22anc 1250 |
. . . . . . . . . . . . . 14
|
| 71 | 68, 70 | breq12d 4034 |
. . . . . . . . . . . . 13
#
# |
| 72 | 10, 14 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 73 | 17, 21 | remulcld 8024 |
. . . . . . . . . . . . . . . 16
|
| 74 | 73 | renegcld 8373 |
. . . . . . . . . . . . . . 15
|
| 75 | 72, 74 | readdcld 8023 |
. . . . . . . . . . . . . 14
|
| 76 | 14, 17 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 77 | 21, 10 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 78 | 76, 77 | readdcld 8023 |
. . . . . . . . . . . . . 14
|
| 79 | 26, 14 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 80 | 30, 21 | remulcld 8024 |
. . . . . . . . . . . . . . . 16
|
| 81 | 80 | renegcld 8373 |
. . . . . . . . . . . . . . 15
|
| 82 | 79, 81 | readdcld 8023 |
. . . . . . . . . . . . . 14
|
| 83 | 14, 30 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 84 | 21, 26 | remulcld 8024 |
. . . . . . . . . . . . . . 15
|
| 85 | 83, 84 | readdcld 8023 |
. . . . . . . . . . . . . 14
|
| 86 | apreim 8596 |
. . . . . . . . . . . . . 14
#
# # | |
| 87 | 75, 78, 82, 85, 86 | syl22anc 1250 |
. . . . . . . . . . . . 13
# # # |
| 88 | 66, 71, 87 | 3bitrd 214 |
. . . . . . . . . . . 12
#
# # |
| 89 | apreim 8596 |
. . . . . . . . . . . . 13
#
#
# | |
| 90 | 10, 17, 26, 30, 89 | syl22anc 1250 |
. . . . . . . . . . . 12
#
#
# |
| 91 | 59, 88, 90 | 3imtr4d 203 |
. . . . . . . . . . 11
# # |
| 92 | 60, 64 | breq12d 4034 |
. . . . . . . . . . 11
#
# |
| 93 | 91, 92 | sylibrd 169 |
. . . . . . . . . 10
# # |
| 94 | 93 | ex 115 |
. . . . . . . . 9
# # |
| 95 | 94 | rexlimdvva 2615 |
. . . . . . . 8
# # |
| 96 | 9, 95 | mpd 13 |
. . . . . . 7
# # |
| 97 | 96 | ex 115 |
. . . . . 6
# # |
| 98 | 97 | rexlimdvva 2615 |
. . . . 5
# # |
| 99 | 5, 98 | mpd 13 |
. . . 4
# # |
| 100 | 99 | ex 115 |
. . 3
# # |
| 101 | 100 | rexlimdvva 2615 |
. 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 1364 wcel 2160 wrex 2469 class class class wbr 4021
(class class class)co 5900 cc 7844 cr 7845 ci 7848 caddc 7849 cmul 7851 cneg 8165 # cap 8574 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8030 df-mnf 8031 df-ltxr 8033 df-sub 8166 df-neg 8167 df-reap 8568 df-ap 8575 |
| This theorem is referenced by: mulext2 8606 mulext 8607 mulap0 8647 apmul1 8781 |
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