Theorem mgcmnt2 33171

Description: The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (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)
mgcmnt2.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mgcmnt2.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
mgcmnt2.3	⊢ (𝜑 → 𝑋 ≲ 𝑌)

Assertion

Ref	Expression
mgcmnt2	⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))

Proof of Theorem mgcmnt2

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

Step	Hyp	Ref	Expression
1		mgcval.3	. . 3 ⊢ (𝜑 → 𝑊 ∈ 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.2	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Proset )
9	3, 4, 5, 6, 7, 8, 1	mgcval 33165	. . . . . 6 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
10	2, 9	mpbid 234	. . . . 5 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦))))
11	10	simplld 777	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	10	simplrd 779	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
13		mgcmnt2.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	12, 13	ffvelcdmd 7066	. . . 4 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
15	11, 14	ffvelcdmd 7066	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ∈ 𝐵)
16		mgcmnt2.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	3, 4, 5, 6, 7, 8, 1, 2, 13	mgccole2 33169	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑋)
18		mgcmnt2.3	. . 3 ⊢ (𝜑 → 𝑋 ≲ 𝑌)
19	4, 6	prstr 18331	. . 3 ⊢ ((𝑊 ∈ Proset ∧ ((𝐹‘(𝐺‘𝑋)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑋 ∧ 𝑋 ≲ 𝑌)) → (𝐹‘(𝐺‘𝑋)) ≲ 𝑌)
20	1, 15, 13, 16, 17, 18, 19	syl132anc 1407	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑌)
21		breq2 5104	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑌))
22		fveq2 6867	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
23	22	breq2d 5112	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐺‘𝑋) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌)))
24	21, 23	bibi12d 347	. . 3 ⊢ (𝑦 = 𝑌 → (((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌))))
25		fveq2 6867	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑋)))
26	25	breq1d 5110	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑋) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑦))
27		breq1 5103	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑋) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))
28	26, 27	bibi12d 347	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦))))
29	28	ralbidv 3185	. . . 4 ⊢ (𝑥 = (𝐺‘𝑋) → (∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦))))
30	10	simprd 499	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
31	29, 30, 14	rspcdva 3582	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))
32	24, 31, 16	rspcdva 3582	. 2 ⊢ (𝜑 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌)))
33	20, 32	mpbid 234	1 ⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   class class class wbr 5100  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  lecple 17293   Proset cproset 18324  MGalConncmgc 33157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-proset 18326  df-mgc 33159

This theorem is referenced by:  dfmgc2  33174  mgcf1olem2  33180