cnmetcoval - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cnmetcoval		Structured version Visualization version GIF version

Theorem cnmetcoval 45947

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 45939	. 2 ⊢ (𝜑 → ((𝐷 ∘ 𝐹)‘𝐵) = ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))))
4	1, 2	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂ × ℂ))
5		xp1st 8016	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (ℂ × ℂ) → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → (1^st ‘(𝐹‘𝐵)) ∈ ℂ)
7		xp2nd 8017	. . . 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 24938	. . 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 595	. 2 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝐵))𝐷(2^nd ‘(𝐹‘𝐵))) = (abs‘((1^st ‘(𝐹‘𝐵)) − (2^nd ‘(𝐹‘𝐵)))))
12	3, 11	eqtrd 2797	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 1569 ∈ wcel 2142 × cxp 5658 ∘ ccom 5664 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 1^st c1st 7982 2^nd c2nd 7983 ℂcc 11104 − cmin 11447 abscabs 15292

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-po 5568 df-so 5569 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-sub 11449

This theorem is used by: (None)

W3C validator