Theorem mgccole2 30780

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 30776	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplrd 770	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
12		mgccole2.1	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	11, 12	ffvelrnd 6836	. . 3 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
14	3, 5	prsref 17593	. . 3 ⊢ ((𝑉 ∈ Proset ∧ (𝐺‘𝑌) ∈ 𝐴) → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
15	1, 13, 14	syl2anc 588	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘𝑌))
16	10	simprd 500	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
17		fveq2 6651	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑌) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑌)))
18	17	breq1d 5035	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑦))
19		breq1 5028	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑌) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)))
20	18, 19	bibi12d 350	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑌) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
21	20	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = (𝐺‘𝑌)) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦))))
22	21	ralbidv 3124	. . . . 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 3531	. . . 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 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
26	25	breq2d 5037	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑌)) ≲ 𝑌))
27		fveq2 6651	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
28	27	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝐺‘𝑦) = (𝐺‘𝑌))
29	28	breq2d 5037	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → ((𝐺‘𝑌) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌)))
30	26, 29	bibi12d 350	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (((𝐹‘(𝐺‘𝑌)) ≲ 𝑦 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑌)) ≲ 𝑌 ↔ (𝐺‘𝑌) ≤ (𝐺‘𝑌))))
31	12, 30	rspcdv 3531	. . 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 260	1 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3068   class class class wbr 5025  ⟶wf 6324  ‘cfv 6328  (class class class)co 7143  Basecbs 16526  lecple 16615   Proset cproset 17587  MGalConncmgc 30768

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-br 5026  df-opab 5088  df-id 5423  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-map 8411  df-proset 17589  df-mgc 30770

This theorem is referenced by:  mgcmnt2  30782  mgcmntco  30783  dfmgc2  30785  mgcf1olem1  30790  mgcf1olem2  30791