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Theorem mgcf1olem1 32773
Description: Property of a Galois connection, lemma for mgcf1o 32775. (Contributed by Thierry Arnoux, 26-Jul-2024.)
Hypotheses
Ref Expression
mgcf1o.h 𝐻 = (𝑉MGalConnπ‘Š)
mgcf1o.a 𝐴 = (Baseβ€˜π‘‰)
mgcf1o.b 𝐡 = (Baseβ€˜π‘Š)
mgcf1o.1 ≀ = (leβ€˜π‘‰)
mgcf1o.2 ≲ = (leβ€˜π‘Š)
mgcf1o.v (πœ‘ β†’ 𝑉 ∈ Poset)
mgcf1o.w (πœ‘ β†’ π‘Š ∈ Poset)
mgcf1o.f (πœ‘ β†’ 𝐹𝐻𝐺)
mgcf1olem1.1 (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
mgcf1olem1 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) = (πΉβ€˜π‘‹))

Proof of Theorem mgcf1olem1
Dummy variables 𝑒 𝑣 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcf1o.w . 2 (πœ‘ β†’ π‘Š ∈ Poset)
2 mgcf1o.f . . . . 5 (πœ‘ β†’ 𝐹𝐻𝐺)
3 mgcf1o.a . . . . . 6 𝐴 = (Baseβ€˜π‘‰)
4 mgcf1o.b . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
5 mgcf1o.1 . . . . . 6 ≀ = (leβ€˜π‘‰)
6 mgcf1o.2 . . . . . 6 ≲ = (leβ€˜π‘Š)
7 mgcf1o.h . . . . . 6 𝐻 = (𝑉MGalConnπ‘Š)
8 mgcf1o.v . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ Poset)
9 posprs 18307 . . . . . . 7 (𝑉 ∈ Poset β†’ 𝑉 ∈ Proset )
108, 9syl 17 . . . . . 6 (πœ‘ β†’ 𝑉 ∈ Proset )
11 posprs 18307 . . . . . . 7 (π‘Š ∈ Poset β†’ π‘Š ∈ Proset )
121, 11syl 17 . . . . . 6 (πœ‘ β†’ π‘Š ∈ Proset )
133, 4, 5, 6, 7, 10, 12dfmgc2 32768 . . . . 5 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 (𝑒 ≲ 𝑣 β†’ (πΊβ€˜π‘’) ≀ (πΊβ€˜π‘£))) ∧ (βˆ€π‘’ ∈ 𝐡 (πΉβ€˜(πΊβ€˜π‘’)) ≲ 𝑒 ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ (πΊβ€˜(πΉβ€˜π‘₯)))))))
142, 13mpbid 231 . . . 4 (πœ‘ β†’ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) ∧ βˆ€π‘’ ∈ 𝐡 βˆ€π‘£ ∈ 𝐡 (𝑒 ≲ 𝑣 β†’ (πΊβ€˜π‘’) ≀ (πΊβ€˜π‘£))) ∧ (βˆ€π‘’ ∈ 𝐡 (πΉβ€˜(πΊβ€˜π‘’)) ≲ 𝑒 ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ (πΊβ€˜(πΉβ€˜π‘₯))))))
1514simplld 766 . . 3 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
1614simplrd 768 . . . 4 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
17 mgcf1olem1.1 . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝐴)
1815, 17ffvelcdmd 7090 . . . 4 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ 𝐡)
1916, 18ffvelcdmd 7090 . . 3 (πœ‘ β†’ (πΊβ€˜(πΉβ€˜π‘‹)) ∈ 𝐴)
2015, 19ffvelcdmd 7090 . 2 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ∈ 𝐡)
213, 4, 5, 6, 7, 10, 12, 2, 18mgccole2 32763 . 2 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ≲ (πΉβ€˜π‘‹))
223, 4, 5, 6, 7, 10, 12, 2, 17mgccole1 32762 . . 3 (πœ‘ β†’ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘‹)))
233, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22mgcmnt1 32764 . 2 (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))))
244, 6posasymb 18310 . . 3 ((π‘Š ∈ Poset ∧ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ∈ 𝐡 ∧ (πΉβ€˜π‘‹) ∈ 𝐡) β†’ (((πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ≲ (πΉβ€˜π‘‹) ∧ (πΉβ€˜π‘‹) ≲ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹)))) ↔ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) = (πΉβ€˜π‘‹)))
2524biimpa 475 . 2 (((π‘Š ∈ Poset ∧ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ∈ 𝐡 ∧ (πΉβ€˜π‘‹) ∈ 𝐡) ∧ ((πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) ≲ (πΉβ€˜π‘‹) ∧ (πΉβ€˜π‘‹) ≲ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))))) β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) = (πΉβ€˜π‘‹))
261, 20, 18, 21, 23, 25syl32anc 1375 1 (πœ‘ β†’ (πΉβ€˜(πΊβ€˜(πΉβ€˜π‘‹))) = (πΉβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051   class class class wbr 5143  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  lecple 17239   Proset cproset 18284  Posetcpo 18298  MGalConncmgc 32751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8845  df-proset 18286  df-poset 18304  df-mgc 32753
This theorem is referenced by:  mgcf1o  32775
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