cnmetcoval - Mathbox for Glauco Siliprandi

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Theorem cnmetcoval 46421

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 46413	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))))
4	1, 2	ffvelcdmd 7109	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ))
5		xp1st 8057	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
7		xp2nd 8058	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (2^nd ‘(𝐹‘𝐵)) ∈ ℂ)
8	4, 7	syl 18	. . 3 ⊢ (𝜑 → (2^nd ‘(𝐹‘𝐵)) ∈ ℂ)
9		cnmetcoval.d	. . . 4 ⊢ 𝐷 = (abs ∘ − )
10	9	cnmetdval 25119	. . 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 596	. 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 × cxp 5653 ∘ ccom 5659 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 1^st c1st 8023 2^nd c2nd 8024 ℂcc 11225 − cmin 11568 abscabs 15430

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8025 df-2nd 8026 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-pnf 11372 df-mnf 11373 df-ltxr 11375 df-sub 11570

This theorem is used by: (None)

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