Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1otrgds Structured version   Visualization version   GIF version

Theorem f1otrgds 25549
 Description: Convenient lemma for f1otrg 25551. (Contributed by Thierry Arnoux, 19-Mar-2019.)
Hypotheses
Ref Expression
f1otrkg.p 𝑃 = (Base‘𝐺)
f1otrkg.d 𝐷 = (dist‘𝐺)
f1otrkg.i 𝐼 = (Itv‘𝐺)
f1otrkg.b 𝐵 = (Base‘𝐻)
f1otrkg.e 𝐸 = (dist‘𝐻)
f1otrkg.j 𝐽 = (Itv‘𝐻)
f1otrkg.f (𝜑𝐹:𝐵1-1-onto𝑃)
f1otrkg.1 ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
f1otrkg.2 ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))
f1otrgitv.x (𝜑𝑋𝐵)
f1otrgitv.y (𝜑𝑌𝐵)
Assertion
Ref Expression
f1otrgds (𝜑 → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌)))
Distinct variable groups:   𝑒,𝑓,𝑔,𝐵   𝐷,𝑒,𝑓   𝑒,𝐸,𝑓   𝑒,𝐹,𝑓,𝑔   𝑒,𝐼,𝑓,𝑔   𝑒,𝐽,𝑓,𝑔   𝑒,𝑋,𝑓,𝑔   𝜑,𝑒,𝑓,𝑔   𝑓,𝑌,𝑔
Allowed substitution hints:   𝐷(𝑔)   𝑃(𝑒,𝑓,𝑔)   𝐸(𝑔)   𝐺(𝑒,𝑓,𝑔)   𝐻(𝑒,𝑓,𝑔)   𝑌(𝑒)

Proof of Theorem f1otrgds
StepHypRef Expression
1 f1otrkg.1 . . 3 ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
21ralrimivva 2954 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 oveq1 6556 . . . . 5 (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6103 . . . . . 6 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
76oveq1d 6564 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑒)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑓)))
85, 7eqeq12d 2625 . . . 4 (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓))))
9 oveq2 6557 . . . . 5 (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10 fveq2 6103 . . . . . 6 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1110oveq2d 6565 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑋)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑌)))
129, 11eqeq12d 2625 . . . 4 (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
138, 12rspc2v 3293 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
143, 4, 13syl2anc 691 . 2 (𝜑 → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
152, 14mpd 15 1 (𝜑 → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  f1otrg  25551
 Copyright terms: Public domain W3C validator