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Theorem dicval 40047
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
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
dicval (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑇,𝑔   𝑓,π‘Š,𝑔,𝑠   𝑓,𝐸,𝑠   𝑃,𝑓   𝑄,𝑓,𝑔,𝑠   𝑇,𝑓
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑔,𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐻(𝑓,𝑔,𝑠)   𝐼(𝑓,𝑔,𝑠)   ≀ (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)
Proof of Theorem dicval
Dummy variables π‘Ÿ π‘ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . . 5 ≀ = (leβ€˜πΎ)
2 dicval.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
3 dicval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 dicval.p . . . . 5 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š)
5 dicval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 dicval.e . . . . 5 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
7 dicval.i . . . . 5 𝐼 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7dicfval 40046 . . . 4 ((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)}))
98adantr 482 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) 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)}))
109fveq1d 6894 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜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)})β€˜π‘„))
11 simpr 486 . . . 4 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
12 breq1 5152 . . . . . 6 (π‘Ÿ = 𝑄 β†’ (π‘Ÿ ≀ π‘Š ↔ 𝑄 ≀ π‘Š))
1312notbid 318 . . . . 5 (π‘Ÿ = 𝑄 β†’ (Β¬ π‘Ÿ ≀ π‘Š ↔ Β¬ 𝑄 ≀ π‘Š))
1413elrab 3684 . . . 4 (𝑄 ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} ↔ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
1511, 14sylibr 233 . . 3 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝑄 ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š})
16 eqeq2 2745 . . . . . . . . 9 (π‘ž = 𝑄 β†’ ((π‘”β€˜π‘ƒ) = π‘ž ↔ (π‘”β€˜π‘ƒ) = 𝑄))
1716riotabidv 7367 . . . . . . . 8 (π‘ž = 𝑄 β†’ (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž) = (℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))
1817fveq2d 6896 . . . . . . 7 (π‘ž = 𝑄 β†’ (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
1918eqeq2d 2744 . . . . . 6 (π‘ž = 𝑄 β†’ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ↔ 𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄))))
2019anbi1d 631 . . . . 5 (π‘ž = 𝑄 β†’ ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)))
2120opabbidv 5215 . . . 4 (π‘ž = 𝑄 β†’ {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = π‘ž)) ∧ 𝑠 ∈ 𝐸)} = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
22 eqid 2733 . . . 4 (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)}) = (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)})
236fvexi 6906 . . . . . . . . . 10 𝐸 ∈ V
2423uniex 7731 . . . . . . . . 9 βˆͺ 𝐸 ∈ V
2524rnex 7903 . . . . . . . 8 ran βˆͺ 𝐸 ∈ V
2625uniex 7731 . . . . . . 7 βˆͺ ran βˆͺ 𝐸 ∈ V
2726pwex 5379 . . . . . 6 𝒫 βˆͺ ran βˆͺ 𝐸 ∈ V
2827, 23xpex 7740 . . . . 5 (𝒫 βˆͺ ran βˆͺ 𝐸 Γ— 𝐸) ∈ V
29 simpl 484 . . . . . . . . 9 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ 𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)))
30 fvssunirn 6925 . . . . . . . . . . 11 (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) βŠ† βˆͺ ran 𝑠
31 elssuni 4942 . . . . . . . . . . . . 13 (𝑠 ∈ 𝐸 β†’ 𝑠 βŠ† βˆͺ 𝐸)
3231adantl 483 . . . . . . . . . . . 12 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ 𝑠 βŠ† βˆͺ 𝐸)
33 rnss 5939 . . . . . . . . . . . 12 (𝑠 βŠ† βˆͺ 𝐸 β†’ ran 𝑠 βŠ† ran βˆͺ 𝐸)
34 uniss 4917 . . . . . . . . . . . 12 (ran 𝑠 βŠ† ran βˆͺ 𝐸 β†’ βˆͺ ran 𝑠 βŠ† βˆͺ ran βˆͺ 𝐸)
3532, 33, 343syl 18 . . . . . . . . . . 11 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ βˆͺ ran 𝑠 βŠ† βˆͺ ran βˆͺ 𝐸)
3630, 35sstrid 3994 . . . . . . . . . 10 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) βŠ† βˆͺ ran βˆͺ 𝐸)
3726elpw2 5346 . . . . . . . . . 10 ((π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∈ 𝒫 βˆͺ ran βˆͺ 𝐸 ↔ (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) βŠ† βˆͺ ran βˆͺ 𝐸)
3836, 37sylibr 233 . . . . . . . . 9 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∈ 𝒫 βˆͺ ran βˆͺ 𝐸)
3929, 38eqeltrd 2834 . . . . . . . 8 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ 𝑓 ∈ 𝒫 βˆͺ ran βˆͺ 𝐸)
40 simpr 486 . . . . . . . 8 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ 𝑠 ∈ 𝐸)
4139, 40jca 513 . . . . . . 7 ((𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸) β†’ (𝑓 ∈ 𝒫 βˆͺ ran βˆͺ 𝐸 ∧ 𝑠 ∈ 𝐸))
4241ssopab2i 5551 . . . . . 6 {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} βŠ† {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 ∈ 𝒫 βˆͺ ran βˆͺ 𝐸 ∧ 𝑠 ∈ 𝐸)}
43 df-xp 5683 . . . . . 6 (𝒫 βˆͺ ran βˆͺ 𝐸 Γ— 𝐸) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 ∈ 𝒫 βˆͺ ran βˆͺ 𝐸 ∧ 𝑠 ∈ 𝐸)}
4442, 43sseqtrri 4020 . . . . 5 {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} βŠ† (𝒫 βˆͺ ran βˆͺ 𝐸 Γ— 𝐸)
4528, 44ssexi 5323 . . . 4 {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ∈ V
4621, 22, 45fvmpt 6999 . . 3 (𝑄 ∈ {π‘Ÿ ∈ 𝐴 ∣ Β¬ π‘Ÿ ≀ π‘Š} β†’ ((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)})β€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
4715, 46syl 17 . 2 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ ((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)})β€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
4810, 47eqtrd 2773 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ (πΌβ€˜π‘„) = {βŸ¨π‘“, π‘ βŸ© ∣ (𝑓 = (π‘ β€˜(℩𝑔 ∈ 𝑇 (π‘”β€˜π‘ƒ) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   Γ— cxp 5675  ran crn 5678  β€˜cfv 6544  β„©crio 7364  lecple 17204  occoc 17205  Atomscatm 38133  LHypclh 38855  LTrncltrn 38972  TEndoctendo 39623  DIsoCcdic 40043
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-dic 40044
This theorem is referenced by:  dicopelval  40048  dicelvalN  40049  dicval2  40050  dicfnN  40054  dicvalrelN  40056  dicssdvh  40057  dicelval1sta  40058  dihpN  40207
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