Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgcf1olem1 Structured version   Visualization version   GIF version

Theorem mgcf1olem1 30856
Description: Property of a Galois connection, lemma for mgcf1o 30858. (Contributed by Thierry Arnoux, 26-Jul-2024.)
Hypotheses
Ref Expression
mgcf1o.h 𝐻 = (𝑉MGalConn𝑊)
mgcf1o.a 𝐴 = (Base‘𝑉)
mgcf1o.b 𝐵 = (Base‘𝑊)
mgcf1o.1 = (le‘𝑉)
mgcf1o.2 = (le‘𝑊)
mgcf1o.v (𝜑𝑉 ∈ Poset)
mgcf1o.w (𝜑𝑊 ∈ Poset)
mgcf1o.f (𝜑𝐹𝐻𝐺)
mgcf1olem1.1 (𝜑𝑋𝐴)
Assertion
Ref Expression
mgcf1olem1 (𝜑 → (𝐹‘(𝐺‘(𝐹𝑋))) = (𝐹𝑋))

Proof of Theorem mgcf1olem1
Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcf1o.w . 2 (𝜑𝑊 ∈ Poset)
2 mgcf1o.f . . . . 5 (𝜑𝐹𝐻𝐺)
3 mgcf1o.a . . . . . 6 𝐴 = (Base‘𝑉)
4 mgcf1o.b . . . . . 6 𝐵 = (Base‘𝑊)
5 mgcf1o.1 . . . . . 6 = (le‘𝑉)
6 mgcf1o.2 . . . . . 6 = (le‘𝑊)
7 mgcf1o.h . . . . . 6 𝐻 = (𝑉MGalConn𝑊)
8 mgcf1o.v . . . . . . 7 (𝜑𝑉 ∈ Poset)
9 posprs 17675 . . . . . . 7 (𝑉 ∈ Poset → 𝑉 ∈ Proset )
108, 9syl 17 . . . . . 6 (𝜑𝑉 ∈ Proset )
11 posprs 17675 . . . . . . 7 (𝑊 ∈ Poset → 𝑊 ∈ Proset )
121, 11syl 17 . . . . . 6 (𝜑𝑊 ∈ Proset )
133, 4, 5, 6, 7, 10, 12dfmgc2 30851 . . . . 5 (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ∧ ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣))) ∧ (∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢 ∧ ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥)))))))
142, 13mpbid 235 . . . 4 (𝜑 → ((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ∧ ∀𝑢𝐵𝑣𝐵 (𝑢 𝑣 → (𝐺𝑢) (𝐺𝑣))) ∧ (∀𝑢𝐵 (𝐹‘(𝐺𝑢)) 𝑢 ∧ ∀𝑥𝐴 𝑥 (𝐺‘(𝐹𝑥))))))
1514simplld 768 . . 3 (𝜑𝐹:𝐴𝐵)
1614simplrd 770 . . . 4 (𝜑𝐺:𝐵𝐴)
17 mgcf1olem1.1 . . . . 5 (𝜑𝑋𝐴)
1815, 17ffvelrnd 6862 . . . 4 (𝜑 → (𝐹𝑋) ∈ 𝐵)
1916, 18ffvelrnd 6862 . . 3 (𝜑 → (𝐺‘(𝐹𝑋)) ∈ 𝐴)
2015, 19ffvelrnd 6862 . 2 (𝜑 → (𝐹‘(𝐺‘(𝐹𝑋))) ∈ 𝐵)
213, 4, 5, 6, 7, 10, 12, 2, 18mgccole2 30846 . 2 (𝜑 → (𝐹‘(𝐺‘(𝐹𝑋))) (𝐹𝑋))
223, 4, 5, 6, 7, 10, 12, 2, 17mgccole1 30845 . . 3 (𝜑𝑋 (𝐺‘(𝐹𝑋)))
233, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22mgcmnt1 30847 . 2 (𝜑 → (𝐹𝑋) (𝐹‘(𝐺‘(𝐹𝑋))))
244, 6posasymb 17678 . . 3 ((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹𝑋))) ∈ 𝐵 ∧ (𝐹𝑋) ∈ 𝐵) → (((𝐹‘(𝐺‘(𝐹𝑋))) (𝐹𝑋) ∧ (𝐹𝑋) (𝐹‘(𝐺‘(𝐹𝑋)))) ↔ (𝐹‘(𝐺‘(𝐹𝑋))) = (𝐹𝑋)))
2524biimpa 480 . 2 (((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹𝑋))) ∈ 𝐵 ∧ (𝐹𝑋) ∈ 𝐵) ∧ ((𝐹‘(𝐺‘(𝐹𝑋))) (𝐹𝑋) ∧ (𝐹𝑋) (𝐹‘(𝐺‘(𝐹𝑋))))) → (𝐹‘(𝐺‘(𝐹𝑋))) = (𝐹𝑋))
261, 20, 18, 21, 23, 25syl32anc 1379 1 (𝜑 → (𝐹‘(𝐺‘(𝐹𝑋))) = (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053   class class class wbr 5030  wf 6335  cfv 6339  (class class class)co 7170  Basecbs 16586  lecple 16675   Proset cproset 17652  Posetcpo 17666  MGalConncmgc 30834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-proset 17654  df-poset 17672  df-mgc 30836
This theorem is referenced by:  mgcf1o  30858
  Copyright terms: Public domain W3C validator