![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 45086 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵)))) |
4 | 1, 2 | ffvelcdmd 7099 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ)) |
5 | xp1st 8039 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹‘𝐵)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘(𝐹‘𝐵)) ∈ ℂ) |
7 | xp2nd 8040 | . . . 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 24788 | . . 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 583 | . 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 1535 ∈ wcel 2104 × cxp 5681 ∘ ccom 5687 ⟶wf 6554 ‘cfv 6558 (class class class)co 7425 1st c1st 8005 2nd c2nd 8006 ℂcc 11144 − cmin 11483 abscabs 15259 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11485 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |