Theorem mgcf1olem1 32976

Description: Property of a Galois connection, lemma for mgcf1o 32978. (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)
mgcf1olem1.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
mgcf1olem1	⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))

Proof of Theorem mgcf1olem1

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

Step	Hyp	Ref	Expression
1		mgcf1o.w	. 2 ⊢ (𝜑 → 𝑊 ∈ 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		mgcf1o.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ Poset)
9		posprs 18374	. . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
10	8, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset )
11		posprs 18374	. . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
12	1, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset )
13	3, 4, 5, 6, 7, 10, 12	dfmgc2 32971	. . . . 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	simplld 768	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
16	14	simplrd 770	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
17		mgcf1olem1.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18	15, 17	ffvelcdmd 7105	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵)
19	16, 18	ffvelcdmd 7105	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑋)) ∈ 𝐴)
20	15, 19	ffvelcdmd 7105	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵)
21	3, 4, 5, 6, 7, 10, 12, 2, 18	mgccole2 32966	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋))
22	3, 4, 5, 6, 7, 10, 12, 2, 17	mgccole1 32965	. . 3 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))
23	3, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22	mgcmnt1 32967	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋))))
24	4, 6	posasymb 18377	. . 3 ⊢ ((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ 𝐵) → (((𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋) ∧ (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋)))) ↔ (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋)))
25	24	biimpa 476	. 2 ⊢ (((𝑊 ∈ Poset ∧ (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ 𝐵) ∧ ((𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋) ∧ (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋))))) → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))
26	1, 20, 18, 21, 23, 25	syl32anc 1377	1 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305   Proset cproset 18350  Posetcpo 18365  MGalConncmgc 32954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-proset 18352  df-poset 18371  df-mgc 32956

This theorem is referenced by:  mgcf1o  32978