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Theorem dicfval 39667
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.)
Hypotheses
Ref Expression
dicval.l ≀ = (leβ€˜πΎ)
dicval.a 𝐴 = (Atomsβ€˜πΎ)
dicval.h 𝐻 = (LHypβ€˜πΎ)
dicval.p 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
dicval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dicval.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dicval.i 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dicfval ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)}))
Distinct variable groups:   𝐴,π‘Ÿ   𝑓,𝑔,π‘ž,π‘Ÿ,𝑠,𝐾   ≀ ,π‘ž   𝐴,π‘ž   𝑇,𝑔   𝑓,π‘Š,𝑔,π‘ž,π‘Ÿ,𝑠
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑓,𝑔,𝑠,π‘Ÿ,π‘ž)   𝑇(𝑓,𝑠,π‘Ÿ,π‘ž)   𝐸(𝑓,𝑔,𝑠,π‘Ÿ,π‘ž)   𝐻(𝑓,𝑔,𝑠,π‘Ÿ,π‘ž)   𝐼(𝑓,𝑔,𝑠,π‘Ÿ,π‘ž)   ≀ (𝑓,𝑔,𝑠,π‘Ÿ)   𝑉(𝑓,𝑔,𝑠,π‘Ÿ,π‘ž)

Proof of Theorem dicfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 dicval.i . . 3 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
2 dicval.l . . . . 5 ≀ = (leβ€˜πΎ)
3 dicval.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
4 dicval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
52, 3, 4dicffval 39666 . . . 4 (𝐾 ∈ 𝑉 β†’ (DIsoCβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))})))
65fveq1d 6849 . . 3 (𝐾 ∈ 𝑉 β†’ ((DIsoCβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))}))β€˜π‘Š))
71, 6eqtrid 2789 . 2 (𝐾 ∈ 𝑉 β†’ 𝐼 = ((𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))}))β€˜π‘Š))
8 breq2 5114 . . . . . 6 (𝑀 = π‘Š β†’ (π‘Ÿ ≀ 𝑀 ↔ π‘Ÿ ≀ π‘Š))
98notbid 318 . . . . 5 (𝑀 = π‘Š β†’ (Β¬ π‘Ÿ ≀ 𝑀 ↔ Β¬ π‘Ÿ ≀ π‘Š))
109rabbidv 3418 . . . 4 (𝑀 = π‘Š β†’ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} = {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š})
11 fveq2 6847 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
12 dicval.t . . . . . . . . . 10 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
1311, 12eqtr4di 2795 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
14 fveq2 6847 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ((ocβ€˜πΎ)β€˜π‘€) = ((ocβ€˜πΎ)β€˜π‘Š))
15 dicval.p . . . . . . . . . . 11 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
1614, 15eqtr4di 2795 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((ocβ€˜πΎ)β€˜π‘€) = 𝑃)
1716fveqeq2d 6855 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž ↔ (π‘”β€˜π‘ƒ) = π‘ž))
1813, 17riotaeqbidv 7321 . . . . . . . 8 (𝑀 = π‘Š β†’ (℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž) = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž))
1918fveq2d 6851 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)))
2019eqeq2d 2748 . . . . . 6 (𝑀 = π‘Š β†’ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ↔ 𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž))))
21 fveq2 6847 . . . . . . . 8 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘Š))
22 dicval.e . . . . . . . 8 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
2321, 22eqtr4di 2795 . . . . . . 7 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
2423eleq2d 2824 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↔ 𝑠 ∈ 𝐸))
2520, 24anbi12d 632 . . . . 5 (𝑀 = π‘Š β†’ ((𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€)) ↔ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)))
2625opabbidv 5176 . . . 4 (𝑀 = π‘Š β†’ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))} = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)})
2710, 26mpteq12dv 5201 . . 3 (𝑀 = π‘Š β†’ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))}) = (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)}))
28 eqid 2737 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))})) = (𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))}))
293fvexi 6861 . . . 4 𝐴 ∈ V
3029mptrabex 7180 . . 3 (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)}) ∈ V
3127, 28, 30fvmpt 6953 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ 𝑀} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘”β€˜((ocβ€˜πΎ)β€˜π‘€)) = π‘ž)) ∧ 𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€))}))β€˜π‘Š) = (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)}))
327, 31sylan9eq 2797 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐼 = (π‘ž ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↦ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3410   class class class wbr 5110  {copab 5172   ↦ cmpt 5193  β€˜cfv 6501  β„©crio 7317  lecple 17147  occoc 17148  Atomscatm 37754  LHypclh 38476  LTrncltrn 38593  TEndoctendo 39244  DIsoCcdic 39664
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-dic 39665
This theorem is referenced by:  dicval  39668  dicfnN  39675
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