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Mirrors > Home > MPE Home > Th. List > subeq0ad | Structured version Visualization version Unicode version |
Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 10307. Generalization of subeq0d 10400. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
negidd.1 | |
pncand.2 |
Ref | Expression |
---|---|
subeq0ad |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 | |
2 | pncand.2 | . 2 | |
3 | subeq0 10307 | . 2 | |
4 | 1, 2, 3 | syl2anc 693 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 cc0 9936 cmin 10266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 |
This theorem is referenced by: subne0ad 10403 subeq0bd 10456 muleqadd 10671 mulcan1g 10680 ofsubeq0 11017 nn0n0n1ge2 11358 mod0 12675 modirr 12741 addmodlteq 12745 sqreulem 14099 sqreu 14100 tanaddlem 14896 fldivp1 15601 4sqlem11 15659 4sqlem16 15664 znf1o 19900 cphsqrtcl2 22986 rrxmet 23191 dvcobr 23709 dvcnvlem 23739 cmvth 23754 dvlip 23756 lhop1lem 23776 ftc1lem5 23803 aalioulem2 24088 sineq0 24273 tanarg 24365 affineequiv 24553 quad2 24566 dcubic 24573 eqeelen 25784 colinearalg 25790 axcontlem7 25850 ipasslem9 27693 ip2eqi 27712 hi2eq 27962 lnopeqi 28867 riesz3i 28921 signslema 30639 circlemeth 30718 poimirlem32 33441 broucube 33443 rrnmet 33628 eqrabdioph 37341 pellexlem1 37393 sineq0ALT 39173 digexp 42401 |
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