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Theorem f1otrgds 27228
Description: Convenient lemma for f1otrg 27230. (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 3111 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 oveq1 7284 . . . . 5 (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6776 . . . . . 6 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
76oveq1d 7292 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑒)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑓)))
85, 7eqeq12d 2754 . . . 4 (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓))))
9 oveq2 7285 . . . . 5 (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10 fveq2 6776 . . . . . 6 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1110oveq2d 7293 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑋)𝐷(𝐹𝑓)) = ((𝐹𝑋)𝐷(𝐹𝑌)))
129, 11eqeq12d 2754 . . . 4 (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹𝑋)𝐷(𝐹𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
138, 12rspc2v 3571 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑒𝐵𝑓𝐵 (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)) → (𝑋𝐸𝑌) = ((𝐹𝑋)𝐷(𝐹𝑌))))
143, 4, 13syl2anc 584 . 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  1-1-ontowf1o 6434  cfv 6435  (class class class)co 7277  Basecbs 16910  distcds 16969  Itvcitv 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-iota 6393  df-fv 6443  df-ov 7280
This theorem is referenced by:  f1otrg  27230
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