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Theorem f1otrgds 26218
Description: Convenient lemma for f1otrg 26220. (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 3153 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 oveq1 6929 . . . . 5 (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6446 . . . . . 6 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
76oveq1d 6937 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑒)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑓)))
85, 7eqeq12d 2793 . . . 4 (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓))))
9 oveq2 6930 . . . . 5 (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10 fveq2 6446 . . . . . 6 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1110oveq2d 6938 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑋)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑌)))
129, 11eqeq12d 2793 . . . 4 (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
138, 12rspc2v 3524 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
143, 4, 13syl2anc 579 . 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  1-1-ontowf1o 6134  cfv 6135  (class class class)co 6922  Basecbs 16255  distcds 16347  Itvcitv 25787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925
This theorem is referenced by:  f1otrg  26220
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