Theorem mgcf1olem1 33085

Description: Property of a Galois connection, lemma for mgcf1o 33087. (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 18241	. . . . . . 7 ⊢ (𝑉 ∈ Poset → 𝑉 ∈ Proset )
10	8, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ Proset )
11		posprs 18241	. . . . . . 7 ⊢ (𝑊 ∈ Poset → 𝑊 ∈ Proset )
12	1, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Proset )
13	3, 4, 5, 6, 7, 10, 12	dfmgc2 33080	. . . . 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 767	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
16	14	simplrd 769	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
17		mgcf1olem1.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18	15, 17	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐵)
19	16, 18	ffvelcdmd 7030	. . 3 ⊢ (𝜑 → (𝐺‘(𝐹‘𝑋)) ∈ 𝐴)
20	15, 19	ffvelcdmd 7030	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ∈ 𝐵)
21	3, 4, 5, 6, 7, 10, 12, 2, 18	mgccole2 33075	. 2 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) ≲ (𝐹‘𝑋))
22	3, 4, 5, 6, 7, 10, 12, 2, 17	mgccole1 33074	. . 3 ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))
23	3, 4, 5, 6, 7, 10, 12, 2, 17, 19, 22	mgcmnt1 33076	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘(𝐺‘(𝐹‘𝑋))))
24	4, 6	posasymb 18244	. . 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 1380	1 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  lecple 17186   Proset cproset 18217  Posetcpo 18232  MGalConncmgc 33063

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767  df-proset 18219  df-poset 18238  df-mgc 33065

This theorem is referenced by:  mgcf1o  33087