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 41475 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵)))) |
4 | 1, 2 | ffvelrnd 6852 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ)) |
5 | xp1st 7721 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹‘𝐵)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘(𝐹‘𝐵)) ∈ ℂ) |
7 | xp2nd 7722 | . . . 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 23379 | . . 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 586 | . 2 ⊢ (𝜑 → ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵))) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
12 | 3, 11 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 × cxp 5553 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 ℂcc 10535 − cmin 10870 abscabs 14593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 |
This theorem is referenced by: (None) |
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