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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnmetcoval | Structured version Visualization version GIF version |
Description: Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
cnmetcoval.d | ⊢ 𝐷 = (abs ∘ − ) |
cnmetcoval.f | ⊢ (𝜑 → 𝐹:𝐴⟶(ℂ × ℂ)) |
cnmetcoval.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
Ref | Expression |
---|---|
cnmetcoval | ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmetcoval.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(ℂ × ℂ)) | |
2 | cnmetcoval.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
3 | 1, 2 | fvovco 39695 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵)))) |
4 | 1, 2 | ffvelrnd 6400 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ)) |
5 | xp1st 7242 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹‘𝐵)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘(𝐹‘𝐵)) ∈ ℂ) |
7 | xp2nd 7243 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹‘𝐵)) ∈ ℂ) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (2nd ‘(𝐹‘𝐵)) ∈ ℂ) |
9 | cnmetcoval.d | . . . 4 ⊢ 𝐷 = (abs ∘ − ) | |
10 | 9 | cnmetdval 22621 | . . 3 ⊢ (((1st ‘(𝐹‘𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹‘𝐵)) ∈ ℂ) → ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵))) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
11 | 6, 8, 10 | syl2anc 694 | . 2 ⊢ (𝜑 → ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵))) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
12 | 3, 11 | eqtrd 2685 | 1 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 × cxp 5141 ∘ ccom 5147 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 ℂcc 9972 − cmin 10304 abscabs 14018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 |
This theorem is referenced by: (None) |
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