ltrnval1 - Mathbox for Norm Megill

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Theorem ltrnval1 35935

Description: Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)

**Hypotheses**
Ref	Expression
ltrnval1.b	⊢ 𝐵 = (Base‘𝐾)
ltrnval1.l	⊢ ≤ = (le‘𝐾)
ltrnval1.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrnval1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
ltrnval1	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

**Proof of Theorem ltrnval1**
Step	Hyp	Ref	Expression
1		ltrnval1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2770	. . . 4 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
3		ltrnval1.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4	1, 2, 3	ltrnldil 35923	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
5	4	3adant3 1125	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6		ltrnval1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
7		ltrnval1.l	. . 3 ⊢ ≤ = (le‘𝐾)
8	6, 7, 1, 2	ldilval 35914	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)
9	5, 8	syld3an2 1517	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 ‘cfv 6031 Basecbs 16063 lecple 16155 LHypclh 35785 LDilcldil 35901 LTrncltrn 35902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pr 5034

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6795 df-ldil 35905 df-ltrn 35906

This theorem is referenced by: ltrnid 35936 ltrnatb 35938 ltrnel 35940 ltrncnvel 35943 ltrneq 35950 ltrnmwOLD 35953 cdlemc2 35994 cdlemd2 36001 cdlemg7N 36428

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