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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 43824 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵)))) |
4 | 1, 2 | ffvelcdmd 7082 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ)) |
5 | xp1st 8001 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹‘𝐵)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘(𝐹‘𝐵)) ∈ ℂ) |
7 | xp2nd 8002 | . . . 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 24268 | . . 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 585 | . 2 ⊢ (𝜑 → ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵))) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
12 | 3, 11 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 × cxp 5672 ∘ ccom 5678 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 1st c1st 7967 2nd c2nd 7968 ℂcc 11103 − cmin 11439 abscabs 15176 |
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 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7969 df-2nd 7970 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-ltxr 11248 df-sub 11441 |
This theorem is referenced by: (None) |
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