Theorem mgcf1olem2 33181

Description: Property of a Galois connection, lemma for mgcf1o 33182. (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 18349	. . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
9	1, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset )
10		mgcf1o.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Poset)
11		posprs 18349	. . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset )
13	3, 4, 5, 6, 7, 9, 12	dfmgc2 33175	. . . . 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 234	. . . 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 779	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
16	14	simplld 777	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
17		mgcf1olem2.1	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	15, 17	ffvelcdmd 7067	. . . 4 ⊢ (𝜑 → (𝐺‘𝑌) ∈ 𝐴)
19	16, 18	ffvelcdmd 7067	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ∈ 𝐵)
20	15, 19	ffvelcdmd 7067	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ∈ 𝐴)
21	3, 4, 5, 6, 7, 9, 12, 2, 17	mgccole2 33170	. . 3 ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
22	3, 4, 5, 6, 7, 9, 12, 2, 19, 17, 21	mgcmnt2 33172	. 2 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) ≤ (𝐺‘𝑌))
23	3, 4, 5, 6, 7, 9, 12, 2, 18	mgccole1 33169	. 2 ⊢ (𝜑 → (𝐺‘𝑌) ≤ (𝐺‘(𝐹‘(𝐺‘𝑌))))
24	3, 5	posasymb 18352	. . 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 480	. 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 1398	1 ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077   class class class wbr 5101  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  lecple 17294   Proset cproset 18325  Posetcpo 18340  MGalConncmgc 33158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-map 8811  df-proset 18327  df-poset 18346  df-mgc 33160

This theorem is referenced by:  mgcf1o  33182