Theorem lautle 40256

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 40255	. . 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 5098	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
7		fveq2 6831	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
8	7	breq1d 5105	. . . 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 5099	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
11		fveq2 6831	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
12	11	breq2d 5107	. . . 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 3584	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  –1-1-onto→wf1o 6488  ‘cfv 6489  Basecbs 17127  lecple 17175  LAutclaut 40157

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 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-map 8761  df-laut 40161

This theorem is referenced by:  lautcnvle  40261  lautlt  40263  lautj  40265  lautm  40266  lauteq  40267  lautco  40269  ltrnle  40301