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Theorem cnmetcoval 45095
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 45086 . 2 (𝜑 → ((𝐷𝐹)‘𝐵) = ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))))
41, 2ffvelcdmd 7099 . . . 4 (𝜑 → (𝐹𝐵) ∈ (ℂ × ℂ))
5 xp1st 8039 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹𝐵)) ∈ ℂ)
64, 5syl 17 . . 3 (𝜑 → (1st ‘(𝐹𝐵)) ∈ ℂ)
7 xp2nd 8040 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹𝐵)) ∈ ℂ)
84, 7syl 17 . . 3 (𝜑 → (2nd ‘(𝐹𝐵)) ∈ ℂ)
9 cnmetcoval.d . . . 4 𝐷 = (abs ∘ − )
109cnmetdval 24788 . . 3 (((1st ‘(𝐹𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹𝐵)) ∈ ℂ) → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
116, 8, 10syl2anc 583 . 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 1535  wcel 2104   × cxp 5681  ccom 5687  wf 6554  cfv 6558  (class class class)co 7425  1st c1st 8005  2nd c2nd 8006  cc 11144  cmin 11483  abscabs 15259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7381  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-er 8738  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11288  df-mnf 11289  df-ltxr 11291  df-sub 11485
This theorem is referenced by: (None)
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