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

Theorem cnmetcoval 39923
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 39912 . 2 (𝜑 → ((𝐷𝐹)‘𝐵) = ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))))
41, 2ffvelrnd 6520 . . . 4 (𝜑 → (𝐹𝐵) ∈ (ℂ × ℂ))
5 xp1st 7368 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹𝐵)) ∈ ℂ)
64, 5syl 17 . . 3 (𝜑 → (1st ‘(𝐹𝐵)) ∈ ℂ)
7 xp2nd 7369 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹𝐵)) ∈ ℂ)
84, 7syl 17 . . 3 (𝜑 → (2nd ‘(𝐹𝐵)) ∈ ℂ)
9 cnmetcoval.d . . . 4 𝐷 = (abs ∘ − )
109cnmetdval 22814 . . 3 (((1st ‘(𝐹𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹𝐵)) ∈ ℂ) → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
116, 8, 10syl2anc 574 . 2 (𝜑 → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
123, 11eqtrd 2808 1 (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1634  wcel 2148   × cxp 5261  ccom 5267  wf 6038  cfv 6042  (class class class)co 6812  1st c1st 7334  2nd c2nd 7335  cc 10157  cmin 10489  abscabs 14204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-po 5184  df-so 5185  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-1st 7336  df-2nd 7337  df-er 7917  df-en 8131  df-dom 8132  df-sdom 8133  df-pnf 10299  df-mnf 10300  df-ltxr 10302  df-sub 10491
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator