Theorem mgcmnt2 32926

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 32920	. . . . . 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	simplld 767	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	10	simplrd 769	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
13		mgcmnt2.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	12, 13	ffvelcdmd 7060	. . . 4 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
15	11, 14	ffvelcdmd 7060	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ∈ 𝐵)
16		mgcmnt2.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	3, 4, 5, 6, 7, 8, 1, 2, 13	mgccole2 32924	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑋)
18		mgcmnt2.3	. . 3 ⊢ (𝜑 → 𝑋 ≲ 𝑌)
19	4, 6	prstr 18267	. . 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 1390	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑋)) ≲ 𝑌)
21		breq2 5114	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑌))
22		fveq2 6861	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
23	22	breq2d 5122	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐺‘𝑋) ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌)))
24	21, 23	bibi12d 345	. . 3 ⊢ (𝑦 = 𝑌 → (((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌))))
25		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑋) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑋)))
26	25	breq1d 5120	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑋) → ((𝐹‘𝑥) ≲ 𝑦 ↔ (𝐹‘(𝐺‘𝑋)) ≲ 𝑦))
27		breq1 5113	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑋) → (𝑥 ≤ (𝐺‘𝑦) ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))
28	26, 27	bibi12d 345	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑋) → (((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)) ↔ ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦))))
29	28	ralbidv 3157	. . . 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 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))
31	29, 30, 14	rspcdva 3592	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ((𝐹‘(𝐺‘𝑋)) ≲ 𝑦 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑦)))
32	24, 31, 16	rspcdva 3592	. 2 ⊢ (𝜑 → ((𝐹‘(𝐺‘𝑋)) ≲ 𝑌 ↔ (𝐺‘𝑋) ≤ (𝐺‘𝑌)))
33	20, 32	mpbid 232	1 ⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234   Proset cproset 18260  MGalConncmgc 32912

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-proset 18262  df-mgc 32914

This theorem is referenced by:  dfmgc2  32929  mgcf1olem2  32935