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Theorem cnmetcoval 45180
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 45171 . 2 (𝜑 → ((𝐷𝐹)‘𝐵) = ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))))
41, 2ffvelcdmd 7103 . . . 4 (𝜑 → (𝐹𝐵) ∈ (ℂ × ℂ))
5 xp1st 8042 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (1st ‘(𝐹𝐵)) ∈ ℂ)
64, 5syl 17 . . 3 (𝜑 → (1st ‘(𝐹𝐵)) ∈ ℂ)
7 xp2nd 8043 . . . 4 ((𝐹𝐵) ∈ (ℂ × ℂ) → (2nd ‘(𝐹𝐵)) ∈ ℂ)
84, 7syl 17 . . 3 (𝜑 → (2nd ‘(𝐹𝐵)) ∈ ℂ)
9 cnmetcoval.d . . . 4 𝐷 = (abs ∘ − )
109cnmetdval 24781 . . 3 (((1st ‘(𝐹𝐵)) ∈ ℂ ∧ (2nd ‘(𝐹𝐵)) ∈ ℂ) → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
116, 8, 10syl2anc 584 . 2 (𝜑 → ((1st ‘(𝐹𝐵))𝐷(2nd ‘(𝐹𝐵))) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
123, 11eqtrd 2776 1 (𝜑 → ((𝐷𝐹)‘𝐵) = (abs‘((1st ‘(𝐹𝐵)) − (2nd ‘(𝐹𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108   × cxp 5681  ccom 5687  wf 6555  cfv 6559  (class class class)co 7429  1st c1st 8008  2nd c2nd 8009  cc 11149  cmin 11488  abscabs 15269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  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 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-er 8741  df-en 8982  df-dom 8983  df-sdom 8984  df-pnf 11293  df-mnf 11294  df-ltxr 11296  df-sub 11490
This theorem is referenced by: (None)
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