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Theorem f1otrgds 28745
Description: Convenient lemma for f1otrg 28747. (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 3190 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 oveq1 7426 . . . . 5 (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6896 . . . . . 6 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
76oveq1d 7434 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑒)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑓)))
85, 7eqeq12d 2741 . . . 4 (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓))))
9 oveq2 7427 . . . . 5 (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10 fveq2 6896 . . . . . 6 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1110oveq2d 7435 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑋)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑌)))
129, 11eqeq12d 2741 . . . 4 (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
138, 12rspc2v 3617 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
143, 4, 13syl2anc 582 . 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  Basecbs 17183  distcds 17245  Itvcitv 28309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  f1otrg  28747
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