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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > absimlere | Structured version Visualization version GIF version |
Description: The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
absimlere.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
absimlere.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
absimlere | ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absimlere.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | absimlere.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 2 | recnd 11184 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
4 | 1, 3 | subcld 11513 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
5 | absimle 15195 | . . 3 ⊢ ((𝐴 − 𝐵) ∈ ℂ → (abs‘(ℑ‘(𝐴 − 𝐵))) ≤ (abs‘(𝐴 − 𝐵))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (abs‘(ℑ‘(𝐴 − 𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
7 | 1, 3 | imsubd 15103 | . . . 4 ⊢ (𝜑 → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
8 | 2 | reim0d 15111 | . . . . 5 ⊢ (𝜑 → (ℑ‘𝐵) = 0) |
9 | 8 | oveq2d 7374 | . . . 4 ⊢ (𝜑 → ((ℑ‘𝐴) − (ℑ‘𝐵)) = ((ℑ‘𝐴) − 0)) |
10 | 1 | imcld 15081 | . . . . . 6 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
11 | 10 | recnd 11184 | . . . . 5 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℂ) |
12 | 11 | subid1d 11502 | . . . 4 ⊢ (𝜑 → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴)) |
13 | 7, 9, 12 | 3eqtrrd 2782 | . . 3 ⊢ (𝜑 → (ℑ‘𝐴) = (ℑ‘(𝐴 − 𝐵))) |
14 | 13 | fveq2d 6847 | . 2 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) = (abs‘(ℑ‘(𝐴 − 𝐵)))) |
15 | 3, 1 | abssubd 15339 | . 2 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (abs‘(𝐴 − 𝐵))) |
16 | 6, 14, 15 | 3brtr4d 5138 | 1 ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 ℂcc 11050 ℝcr 11051 0cc0 11052 ≤ cle 11191 − cmin 11386 ℑcim 14984 abscabs 15120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 |
This theorem is referenced by: cnrefiisplem 44077 |
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