Theorem mgcf1olem2 32983

Description: Property of a Galois connection, lemma for mgcf1o 32984. (Contributed by Thierry Arnoux, 26-Jul-2024.)

Hypotheses

Ref	Expression
mgcf1o.h	⊢ 𝐻 = (𝑉MGalConn𝑊)
mgcf1o.a	⊢ 𝐴 = (Base‘𝑉)
mgcf1o.b	⊢ 𝐵 = (Base‘𝑊)
mgcf1o.1	⊢ ≤ = (le‘𝑉)
mgcf1o.2	⊢ ≲ = (le‘𝑊)
mgcf1o.v	⊢ (𝜑 → 𝑉 ∈ Poset)
mgcf1o.w	⊢ (𝜑 → 𝑊 ∈ Poset)
mgcf1o.f	⊢ (𝜑 → 𝐹𝐻𝐺)
mgcf1olem2.1	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
mgcf1olem2	⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))

Proof of Theorem mgcf1olem2

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

Step	Hyp	Ref	Expression
1		mgcf1o.v	. 2 ⊢ (𝜑 → 𝑉 ∈ Poset)
2		mgcf1o.f	. . . . 5 ⊢ (𝜑 → 𝐹𝐻𝐺)
3		mgcf1o.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
4		mgcf1o.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
5		mgcf1o.1	. . . . . 6 ⊢ ≤ = (le‘𝑉)
6		mgcf1o.2	. . . . . 6 ⊢ ≲ = (le‘𝑊)
7		mgcf1o.h	. . . . . 6 ⊢ 𝐻 = (𝑉MGalConn𝑊)
8		posprs 18222	. . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
9	1, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset )
10		mgcf1o.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset)
11		posprs 18222	. . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset )
13	3, 4, 5, 6, 7, 9, 12	dfmgc2 32977	. . . . 5 ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))
14	2, 13	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥))))))
15	14	simplrd 769	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
16	14	simplld 767	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
17		mgcf1olem2.1	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	15, 17	ffvelcdmd 7018	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
19	16, 18	ffvelcdmd 7018	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ∈ 𝐵)
20	15, 19	ffvelcdmd 7018	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴)
21	3, 4, 5, 6, 7, 9, 12, 2, 17	mgccole2 32972	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
22	3, 4, 5, 6, 7, 9, 12, 2, 19, 17, 21	mgcmnt2 32974	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌))
23	3, 4, 5, 6, 7, 9, 12, 2, 18	mgccole1 32971	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌))))
24	3, 5	posasymb 18225	. . 3 ⊢ ((𝑉 ∈ Poset ∧ (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴 ∧ (𝐺‘𝑌) ∈ 𝐴) → (((𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌) ∧ (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌)))) ↔ (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌)))
25	24	biimpa 476	. 2 ⊢ (((𝑉 ∈ Poset ∧ (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴 ∧ (𝐺‘𝑌) ∈ 𝐴) ∧ ((𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌) ∧ (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌))))) → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))
26	1, 20, 18, 22, 23, 25	syl32anc 1380	1 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5089  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168   Proset cproset 18198  Posetcpo 18213  MGalConncmgc 32960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-proset 18200  df-poset 18219  df-mgc 32962

This theorem is referenced by:  mgcf1o  32984