Theorem mgcf1olem2 33080

Description: Property of a Galois connection, lemma for mgcf1o 33081. (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 18276	. . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
9	1, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset )
10		mgcf1o.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset)
11		posprs 18276	. . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset )
13	3, 4, 5, 6, 7, 9, 12	dfmgc2 33074	. . . . 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 770	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
16	14	simplld 768	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
17		mgcf1olem2.1	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	15, 17	ffvelcdmd 7032	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
19	16, 18	ffvelcdmd 7032	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ∈ 𝐵)
20	15, 19	ffvelcdmd 7032	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴)
21	3, 4, 5, 6, 7, 9, 12, 2, 17	mgccole2 33069	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
22	3, 4, 5, 6, 7, 9, 12, 2, 19, 17, 21	mgcmnt2 33071	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌))
23	3, 4, 5, 6, 7, 9, 12, 2, 18	mgccole1 33068	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌))))
24	3, 5	posasymb 18279	. . 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 1381	1 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221   Proset cproset 18252  Posetcpo 18267  MGalConncmgc 33057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8769  df-proset 18254  df-poset 18273  df-mgc 33059

This theorem is referenced by:  mgcf1o  33081