Theorem mgccole2 32971

Description: Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.)

Hypotheses

Ref	Expression
mgcoval.1	⊢ 𝐴 = (Base‘𝑉)
mgcoval.2	⊢ 𝐵 = (Base‘𝑊)
mgcoval.3	⊢ ≤ = (le‘𝑉)
mgcoval.4	⊢ ≲ = (le‘𝑊)
mgcval.1	⊢ 𝐻 = (𝑉MGalConn𝑊)
mgcval.2	⊢ (𝜑 → 𝑉 ∈ Proset )
mgcval.3	⊢ (𝜑 → 𝑊 ∈ Proset )
mgccole.1	⊢ (𝜑 → 𝐹𝐻𝐺)
mgccole2.1	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
mgccole2	⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

Proof of Theorem mgccole2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgcval.2	. . 3 ⊢ (𝜑 → 𝑉 ∈ Proset )
2		mgccole.1	. . . . . 6 ⊢ (𝜑 → 𝐹𝐻𝐺)
3		mgcoval.1	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑉)
4		mgcoval.2	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
5		mgcoval.3	. . . . . . 7 ⊢ ≤ = (le‘𝑉)
6		mgcoval.4	. . . . . . 7 ⊢ ≲ = (le‘𝑊)
7		mgcval.1	. . . . . . 7 ⊢ 𝐻 = (𝑉MGalConn𝑊)
8		mgcval.3	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Proset )
9	3, 4, 5, 6, 7, 1, 8	mgcval 32967	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 232	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplrd 769	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12		mgccole2.1	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	11, 12	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
14	3, 5	prsref 18310	. . 3 ⊢ ((𝑉 ∈ Proset ∧ (𝐺‘𝑌) ∈ 𝐴) → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
15	1, 13, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
16	10	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
17		fveq2 6876	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑌) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑌)))
18	17	breq1d 5129	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑦))
19		breq1 5122	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
20	18, 19	bibi12d 345	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑌) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
21	20	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
22	21	ralbidv 3163	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
23	13, 22	rspcdv 3593	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
24	16, 23	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
25		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
26	25	breq2d 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑌))
27		fveq2 6876	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
28	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝐺‘𝑦) = (𝐺‘𝑌))
29	28	breq2d 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐺‘𝑌) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
30	26, 29	bibi12d 345	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
31	12, 30	rspcdv 3593	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
32	24, 31	mpd 15	. 2 ⊢ (𝜑 → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
33	15, 32	mpbird 257	1 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   class class class wbr 5119  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278   Proset cproset 18304  MGalConncmgc 32959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8842  df-proset 18306  df-mgc 32961

This theorem is referenced by:  mgcmnt2  32973  mgcmntco  32974  dfmgc2  32976  mgcf1olem1  32981  mgcf1olem2  32982