cnmetcoval - Mathbox for Glauco Siliprandi

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Theorem cnmetcoval 45304

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 45295	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))))
4	1, 2	ffvelcdmd 7024	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ))
5		xp1st 7959	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
7		xp2nd 7960	. . . 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 24691	. . 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 584	. 2 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))
12	3, 11	eqtrd 2766	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 1541 ∈ wcel 2111 × cxp 5617 ∘ ccom 5623 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 ℂcc 11010 − cmin 11350 abscabs 15147

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11154 df-mnf 11155 df-ltxr 11157 df-sub 11352

This theorem is referenced by: (None)

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