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Mirrors > Home > MPE Home > Th. List > subeq0ad | Structured version Visualization version GIF version |
Description: The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 10649. Generalization of subeq0d 10742. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
subeq0ad | ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subeq0 10649 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
4 | 1, 2, 3 | syl2anc 579 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 0cc0 10272 − cmin 10606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 |
This theorem is referenced by: subne0ad 10745 subeq0bd 10801 muleqadd 11019 mulcan1g 11028 ofsubeq0 11371 nn0n0n1ge2 11709 mod0 12994 modirr 13060 addmodlteq 13064 sqreulem 14506 sqreu 14507 tanaddlem 15298 fldivp1 16005 4sqlem11 16063 4sqlem16 16068 znf1o 20295 cphsqrtcl2 23393 rrxmet 23614 dvcobr 24146 dvcnvlem 24176 cmvth 24191 dvlip 24193 lhop1lem 24213 ftc1lem5 24240 aalioulem2 24525 sineq0 24711 tanarg 24802 affineequiv 25001 quad2 25017 dcubic 25024 eqeelen 26253 colinearalg 26259 axcontlem7 26319 ipasslem9 28265 ip2eqi 28284 hi2eq 28534 lnopeqi 29439 riesz3i 29493 signslema 31239 circlemeth 31320 poimirlem32 34069 broucube 34071 rrnmet 34254 eqrabdioph 38305 pellexlem1 38357 sineq0ALT 40110 digexp 43420 eenglngeehlnmlem2 43478 2itscp 43521 |
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