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Theorem cnmetcoval 43833
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 43824 . 2 (𝜑 → ((𝐷𝐹)‘𝐵) = ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))))
41, 2ffvelcdmd 7082 . . . 4 (𝜑 → (𝐹𝐵) ∈ (ℂ × ℂ))
5 xp1st 8001 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹𝐵)) ∈ ℂ)
64, 5syl 17 . . 3 (𝜑 → (1st ‘(𝐹𝐵)) ∈ ℂ)
7 xp2nd 8002 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹𝐵)) ∈ ℂ)
84, 7syl 17 . . 3 (𝜑 → (2nd ‘(𝐹𝐵)) ∈ ℂ)
9 cnmetcoval.d . . . 4 𝐷 = (abs ∘ − )
109cnmetdval 24268 . . 3 (((1st ‘(𝐹𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹𝐵)) ∈ ℂ) → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
116, 8, 10syl2anc 585 . 2 (𝜑 → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
123, 11eqtrd 2773 1 (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   × cxp 5672  ccom 5678  wf 6535  cfv 6539  (class class class)co 7403  1st c1st 7967  2nd c2nd 7968  cc 11103  cmin 11439  abscabs 15176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-po 5586  df-so 5587  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7969  df-2nd 7970  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11245  df-mnf 11246  df-ltxr 11248  df-sub 11441
This theorem is referenced by: (None)
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