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Mirrors > Home > MPE Home > Th. List > absnegd | Structured version Visualization version GIF version |
Description: Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
absnegd | ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | absneg 15250 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 ℂcc 11130 -cneg 11469 abscabs 15207 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-2 12299 df-cj 15072 df-re 15073 df-im 15074 df-abs 15209 |
This theorem is referenced by: bddiblnc 25764 abelthlem8 26369 tanabsge 26434 abssinper 26448 cxpcn3 26676 abscxpbnd 26681 cosangneg2d 26732 chordthmlem 26757 atantayl 26862 lgamgulmlem2 26955 lgambdd 26962 lgsneg 27247 pntibndlem2 27517 irrdiff 36795 poimirlem29 37111 ftc1anclem8 37162 binomcxplemnotnn0 43765 neglimc 45007 stirlinglem5 45438 fourierdlem30 45497 fourierdlem39 45506 fourierdlem47 45513 fourierdlem73 45539 etransclem41 45635 hoicvr 45908 |
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