Theorem lautle 39551

Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Hypotheses

Ref	Expression
lautset.b	⊢ 𝐵 = (Base‘𝐾)
lautset.l	⊢ ≤ = (le‘𝐾)
lautset.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautle	⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

Proof of Theorem lautle

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

Step	Hyp	Ref	Expression
1		lautset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lautset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		lautset.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
4	1, 2, 3	islaut 39550	. . 3 ⊢ (𝐾 ∈ 𝑉 → (𝐹 ∈ 𝐼 ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))))
5	4	simplbda 499	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
6		breq1 5145	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
7		fveq2 6891	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
8	7	breq1d 5152	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)))
9	6, 8	bibi12d 345	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦))))
10		breq2 5146	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
11		fveq2 6891	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
12	11	breq2d 5154	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))
13	10, 12	bibi12d 345	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))))
14	9, 13	rspc2v 3619	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))))
15	5, 14	mpan9 506	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3057   class class class wbr 5142  –1-1-onto→wf1o 6541  ‘cfv 6542  Basecbs 17173  lecple 17233  LAutclaut 39452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-laut 39456

This theorem is referenced by:  lautcnvle  39556  lautlt  39558  lautj  39560  lautm  39561  lauteq  39562  lautco  39564  ltrnle  39596