cnmetcoval - Mathbox for Glauco Siliprandi

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Theorem cnmetcoval 45770

Description: Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
cnmetcoval.d	⊢ 𝐷 = (abs ∘ − )
cnmetcoval.f	⊢ (𝜑 → 𝐹:𝐴⟶(ℂ × ℂ))
cnmetcoval.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)

**Assertion**
Ref	Expression
cnmetcoval	⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))

**Proof of Theorem cnmetcoval**
Step	Hyp	Ref	Expression
1		cnmetcoval.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(ℂ × ℂ))
2		cnmetcoval.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
3	1, 2	fvovco 45762	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))))
4	1, 2	ffvelcdmd 7066	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ))
5		xp1st 8002	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
7		xp2nd 8003	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (2^nd ‘(𝐹‘𝐵)) ∈ ℂ)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (2^nd ‘(𝐹‘𝐵)) ∈ ℂ)
9		cnmetcoval.d	. . . 4 ⊢ 𝐷 = (abs ∘ − )
10	9	cnmetdval 24837	. . 3 ⊢ (((1^st ‘(𝐹‘𝐵)) ∈ ℂ ∧ (2^nd ‘(𝐹‘𝐵)) ∈ ℂ) → ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))
11	6, 8, 10	syl2anc 593	. 2 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))
12	3, 11	eqtrd 2798	1 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1561 ∈ wcel 2143 × cxp 5646 ∘ ccom 5652 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 ℂcc 11082 − cmin 11425 abscabs 15271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-id 5543 df-po 5556 df-so 5557 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11229 df-mnf 11230 df-ltxr 11232 df-sub 11427

This theorem is referenced by: (None)

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