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Theorem f1otrgds 25440
Description: Convenient lemma for f1otrg 25442. (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 2858 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 oveq1 6432 . . . . 5 (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 5986 . . . . . 6 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
76oveq1d 6440 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑒)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑓)))
85, 7eqeq12d 2529 . . . 4 (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓))))
9 oveq2 6433 . . . . 5 (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10 fveq2 5986 . . . . . 6 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1110oveq2d 6441 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑋)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑌)))
129, 11eqeq12d 2529 . . . 4 (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
138, 12rspc2v 3197 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
143, 4, 13syl2anc 690 . 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  1-1-ontowf1o 5688  cfv 5689  (class class class)co 6425  Basecbs 15577  distcds 15659  Itvcitv 25025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-iota 5653  df-fv 5697  df-ov 6428
This theorem is referenced by:  f1otrg  25442
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