![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11538. Generalization of subeq0d 11631. (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 11538 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 (class class class)co 7426 ℂcc 11158 0cc0 11160 − cmin 11496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-ltxr 11305 df-sub 11498 |
This theorem is referenced by: subne0ad 11634 subeq0bd 11692 muleqadd 11910 mulcan1g 11919 ofsubeq0 12263 nn0n0n1ge2 12593 mod0 13898 modirr 13964 addmodlteq 13968 sqreulem 15366 sqreu 15367 tanaddlem 16170 fldivp1 16901 4sqlem11 16959 4sqlem16 16964 znf1o 21551 cphsqrtcl2 25208 rrxmet 25430 dvcobr 25971 dvcobrOLD 25972 dvcnvlem 26002 cmvth 26017 cmvthOLD 26018 dvlip 26020 lhop1lem 26040 ftc1lem5 26069 aalioulem2 26364 sineq0 26554 tanarg 26649 affineequiv 26854 quad2 26870 dcubic 26877 eqeelen 28841 colinearalg 28847 axcontlem7 28907 ipasslem9 30774 ip2eqi 30792 hi2eq 31041 lnopeqi 31944 riesz3i 31998 2sqr3minply 33609 signslema 34410 circlemeth 34488 poimirlem32 37355 broucube 37357 rrnmet 37532 eqrabdioph 42452 pellexlem1 42504 sineq0ALT 44631 digexp 48013 eenglngeehlnmlem2 48144 2itscp 48187 |
Copyright terms: Public domain | W3C validator |