cnmetcoval - Mathbox for Glauco Siliprandi

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Theorem cnmetcoval 42631

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 42621	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))))
4	1, 2	ffvelrnd 6944	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ))
5		xp1st 7836	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
7		xp2nd 7837	. . . 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 23840	. . 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 583	. 2 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))
12	3, 11	eqtrd 2778	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 1539 ∈ wcel 2108 × cxp 5578 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 ℂcc 10800 − cmin 11135 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-ltxr 10945 df-sub 11137

This theorem is referenced by: (None)

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