Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgcoval Structured version   Visualization version   GIF version

Theorem mgcoval 30678
 Description: Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Base‘𝑉)
mgcoval.2 𝐵 = (Base‘𝑊)
mgcoval.3 = (le‘𝑉)
mgcoval.4 = (le‘𝑊)
Assertion
Ref Expression
mgcoval ((𝑉𝑋𝑊𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦   𝐵,𝑓,𝑔,𝑥,𝑦   𝑓,𝑉,𝑔,𝑥,𝑦   𝑓,𝑊,𝑔,𝑥,𝑦   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,𝑔,𝑥,𝑦
Allowed substitution hints:   (𝑥,𝑦,𝑓,𝑔)   (𝑥,𝑦,𝑓,𝑔)

Proof of Theorem mgcoval
Dummy variables 𝑎 𝑏 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mgc 30673 . . 3 MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))})
21a1i 11 . 2 ((𝑉𝑋𝑊𝑌) → MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))}))
3 fvexd 6667 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) ∈ V)
4 simprl 770 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → 𝑣 = 𝑉)
54fveq2d 6656 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = (Base‘𝑉))
6 mgcoval.1 . . . 4 𝐴 = (Base‘𝑉)
75, 6eqtr4di 2875 . . 3 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) = 𝐴)
8 fvexd 6667 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) ∈ V)
9 simplrr 777 . . . . . 6 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
109fveq2d 6656 . . . . 5 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
11 mgcoval.2 . . . . 5 𝐵 = (Base‘𝑊)
1210, 11eqtr4di 2875 . . . 4 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
13 simpr 488 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14 simplr 768 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴)
1513, 14oveq12d 7158 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑏m 𝑎) = (𝐵m 𝐴))
1615eleq2d 2899 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝑏m 𝑎) ↔ 𝑓 ∈ (𝐵m 𝐴)))
1714, 13oveq12d 7158 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑎m 𝑏) = (𝐴m 𝐵))
1817eleq2d 2899 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑔 ∈ (𝑎m 𝑏) ↔ 𝑔 ∈ (𝐴m 𝐵)))
1916, 18anbi12d 633 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ↔ (𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵))))
209adantr 484 . . . . . . . . . . . 12 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑤 = 𝑊)
2120fveq2d 6656 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = (le‘𝑊))
22 mgcoval.4 . . . . . . . . . . 11 = (le‘𝑊)
2321, 22eqtr4di 2875 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑤) = )
2423breqd 5053 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → ((𝑓𝑥)(le‘𝑤)𝑦 ↔ (𝑓𝑥) 𝑦))
254ad2antrr 725 . . . . . . . . . . . 12 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → 𝑣 = 𝑉)
2625fveq2d 6656 . . . . . . . . . . 11 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = (le‘𝑉))
27 mgcoval.3 . . . . . . . . . . 11 = (le‘𝑉)
2826, 27eqtr4di 2875 . . . . . . . . . 10 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (le‘𝑣) = )
2928breqd 5053 . . . . . . . . 9 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (𝑥(le‘𝑣)(𝑔𝑦) ↔ 𝑥 (𝑔𝑦)))
3024, 29bibi12d 349 . . . . . . . 8 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3113, 30raleqbidv 3382 . . . . . . 7 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3214, 31raleqbidv 3382 . . . . . 6 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦))))
3319, 32anbi12d 633 . . . . 5 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → (((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦))) ↔ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))))
3433opabbidv 5108 . . . 4 (((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) ∧ 𝑏 = 𝐵) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
358, 12, 34csbied2 3892 . . 3 ((((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) ∧ 𝑎 = 𝐴) → (Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
363, 7, 35csbied2 3892 . 2 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) / 𝑎(Base‘𝑤) / 𝑏{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏m 𝑎) ∧ 𝑔 ∈ (𝑎m 𝑏)) ∧ ∀𝑥𝑎𝑦𝑏 ((𝑓𝑥)(le‘𝑤)𝑦𝑥(le‘𝑣)(𝑔𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
37 simpl 486 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑉𝑋)
3837elexd 3489 . 2 ((𝑉𝑋𝑊𝑌) → 𝑉 ∈ V)
39 simpr 488 . . 3 ((𝑉𝑋𝑊𝑌) → 𝑊𝑌)
4039elexd 3489 . 2 ((𝑉𝑋𝑊𝑌) → 𝑊 ∈ V)
41 ovexd 7175 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐵m 𝐴) ∈ V)
42 ovexd 7175 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐴m 𝐵) ∈ V)
43 simprll 778 . . 3 (((𝑉𝑋𝑊𝑌) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))) → 𝑓 ∈ (𝐵m 𝐴))
44 simprlr 779 . . 3 (((𝑉𝑋𝑊𝑌) ∧ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))) → 𝑔 ∈ (𝐴m 𝐵))
4541, 42, 43, 44opabex2 7741 . 2 ((𝑉𝑋𝑊𝑌) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))} ∈ V)
462, 36, 38, 40, 45ovmpod 7286 1 ((𝑉𝑋𝑊𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵m 𝐴) ∧ 𝑔 ∈ (𝐴m 𝐵)) ∧ ∀𝑥𝐴𝑦𝐵 ((𝑓𝑥) 𝑦𝑥 (𝑔𝑦)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  Vcvv 3469  ⦋csb 3855   class class class wbr 5042  {copab 5104  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142   ↑m cmap 8393  Basecbs 16474  lecple 16563  MGalConncmgc 30671 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-mgc 30673 This theorem is referenced by:  mgcval  30679
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