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 38376 | . 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵)))) |
4 | 1, 2 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ)) |
5 | xp1st 7089 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹‘𝐵)) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (1st ‘(𝐹‘𝐵)) ∈ ℂ) |
7 | xp2nd 7090 | . . . 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 22384 | . . 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 691 | . 2 ⊢ (𝜑 → ((1st ‘(𝐹‘𝐵))𝐷(2nd ‘(𝐹‘𝐵))) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
12 | 3, 11 | eqtrd 2644 | 1 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1st ‘(𝐹‘𝐵)) − (2nd ‘(𝐹‘𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 ℂcc 9813 − cmin 10145 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 |
This theorem is referenced by: (None) |
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