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Theorem cnmetcoval 39708
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 (𝜑𝐹:𝐴⟶(ℂ × ℂ))
cnmetcoval.b (𝜑𝐵𝐴)
Assertion
Ref Expression
cnmetcoval (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))

Proof of Theorem cnmetcoval
StepHypRef Expression
1 cnmetcoval.f . . 3 (𝜑𝐹:𝐴⟶(ℂ × ℂ))
2 cnmetcoval.b . . 3 (𝜑𝐵𝐴)
31, 2fvovco 39695 . 2 (𝜑 → ((𝐷𝐹)‘𝐵) = ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))))
41, 2ffvelrnd 6400 . . . 4 (𝜑 → (𝐹𝐵) ∈ (ℂ × ℂ))
5 xp1st 7242 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹𝐵)) ∈ ℂ)
64, 5syl 17 . . 3 (𝜑 → (1st ‘(𝐹𝐵)) ∈ ℂ)
7 xp2nd 7243 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹𝐵)) ∈ ℂ)
84, 7syl 17 . . 3 (𝜑 → (2nd ‘(𝐹𝐵)) ∈ ℂ)
9 cnmetcoval.d . . . 4 𝐷 = (abs ∘ − )
109cnmetdval 22621 . . 3 (((1st ‘(𝐹𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹𝐵)) ∈ ℂ) → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
116, 8, 10syl2anc 694 . 2 (𝜑 → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
123, 11eqtrd 2685 1 (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  cc 9972  cmin 10304  abscabs 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-sub 10306
This theorem is referenced by: (None)
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