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Theorem mgcmnt1 31901
Description: The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
Hypotheses
Ref Expression
mgcoval.1 𝐴 = (Baseβ€˜π‘‰)
mgcoval.2 𝐡 = (Baseβ€˜π‘Š)
mgcoval.3 ≀ = (leβ€˜π‘‰)
mgcoval.4 ≲ = (leβ€˜π‘Š)
mgcval.1 𝐻 = (𝑉MGalConnπ‘Š)
mgcval.2 (πœ‘ β†’ 𝑉 ∈ Proset )
mgcval.3 (πœ‘ β†’ π‘Š ∈ Proset )
mgccole.1 (πœ‘ β†’ 𝐹𝐻𝐺)
mgcmnt1.1 (πœ‘ β†’ 𝑋 ∈ 𝐴)
mgcmnt1.2 (πœ‘ β†’ π‘Œ ∈ 𝐴)
mgcmnt1.3 (πœ‘ β†’ 𝑋 ≀ π‘Œ)
Assertion
Ref Expression
mgcmnt1 (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))

Proof of Theorem mgcmnt1
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgcval.2 . . 3 (πœ‘ β†’ 𝑉 ∈ Proset )
2 mgcmnt1.1 . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐴)
3 mgcmnt1.2 . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐴)
4 mgccole.1 . . . . . 6 (πœ‘ β†’ 𝐹𝐻𝐺)
5 mgcoval.1 . . . . . . 7 𝐴 = (Baseβ€˜π‘‰)
6 mgcoval.2 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
7 mgcoval.3 . . . . . . 7 ≀ = (leβ€˜π‘‰)
8 mgcoval.4 . . . . . . 7 ≲ = (leβ€˜π‘Š)
9 mgcval.1 . . . . . . 7 𝐻 = (𝑉MGalConnπ‘Š)
10 mgcval.3 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Proset )
115, 6, 7, 8, 9, 1, 10mgcval 31896 . . . . . 6 (πœ‘ β†’ (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))))
124, 11mpbid 231 . . . . 5 (πœ‘ β†’ ((𝐹:𝐴⟢𝐡 ∧ 𝐺:𝐡⟢𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦))))
1312simplrd 769 . . . 4 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
1412simplld 767 . . . . 5 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
1514, 3ffvelcdmd 7037 . . . 4 (πœ‘ β†’ (πΉβ€˜π‘Œ) ∈ 𝐡)
1613, 15ffvelcdmd 7037 . . 3 (πœ‘ β†’ (πΊβ€˜(πΉβ€˜π‘Œ)) ∈ 𝐴)
17 mgcmnt1.3 . . 3 (πœ‘ β†’ 𝑋 ≀ π‘Œ)
185, 6, 7, 8, 9, 1, 10, 4, 3mgccole1 31899 . . 3 (πœ‘ β†’ π‘Œ ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))
195, 7prstr 18194 . . 3 ((𝑉 ∈ Proset ∧ (𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐴 ∧ (πΊβ€˜(πΉβ€˜π‘Œ)) ∈ 𝐴) ∧ (𝑋 ≀ π‘Œ ∧ π‘Œ ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))) β†’ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))
201, 2, 3, 16, 17, 18, 19syl132anc 1389 . 2 (πœ‘ β†’ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))
2112simprd 497 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)))
22 fveq2 6843 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
2322breq1d 5116 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ (πΉβ€˜π‘‹) ≲ 𝑦))
24 breq1 5109 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ (πΊβ€˜π‘¦) ↔ 𝑋 ≀ (πΊβ€˜π‘¦)))
2523, 24bibi12d 346 . . . . . . 7 (π‘₯ = 𝑋 β†’ (((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦))))
2625adantl 483 . . . . . 6 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ (((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦))))
2726ralbidv 3171 . . . . 5 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦))))
282, 27rspcdv 3572 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘₯) ≲ 𝑦 ↔ π‘₯ ≀ (πΊβ€˜π‘¦)) β†’ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦))))
2921, 28mpd 15 . . 3 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦)))
30 simpr 486 . . . . . 6 ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘Œ)) β†’ 𝑦 = (πΉβ€˜π‘Œ))
3130breq2d 5118 . . . . 5 ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘Œ)) β†’ ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ)))
3230fveq2d 6847 . . . . . 6 ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘Œ)) β†’ (πΊβ€˜π‘¦) = (πΊβ€˜(πΉβ€˜π‘Œ)))
3332breq2d 5118 . . . . 5 ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘Œ)) β†’ (𝑋 ≀ (πΊβ€˜π‘¦) ↔ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ))))
3431, 33bibi12d 346 . . . 4 ((πœ‘ ∧ 𝑦 = (πΉβ€˜π‘Œ)) β†’ (((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦)) ↔ ((πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ) ↔ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))))
3515, 34rspcdv 3572 . . 3 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜π‘‹) ≲ 𝑦 ↔ 𝑋 ≀ (πΊβ€˜π‘¦)) β†’ ((πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ) ↔ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ)))))
3629, 35mpd 15 . 2 (πœ‘ β†’ ((πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ) ↔ 𝑋 ≀ (πΊβ€˜(πΉβ€˜π‘Œ))))
3720, 36mpbird 257 1 (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145   Proset cproset 18187  MGalConncmgc 31888
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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-proset 18189  df-mgc 31890
This theorem is referenced by:  dfmgc2  31905  mgcf1olem1  31910
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