Theorem istrnN 40820

Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
trnset.a	⊢ 𝐴 = (Atoms‘𝐾)
trnset.s	⊢ 𝑆 = (PSubSp‘𝐾)
trnset.p	⊢ + = (+𝑃‘𝐾)
trnset.o	⊢ ⊥ = (⊥𝑃‘𝐾)
trnset.w	⊢ 𝑊 = (WAtoms‘𝐾)
trnset.m	⊢ 𝑀 = (PAut‘𝐾)
trnset.l	⊢ 𝐿 = (Dil‘𝐾)
trnset.t	⊢ 𝑇 = (Trn‘𝐾)

Assertion

Ref	Expression
istrnN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))))

Distinct variable groups:   𝑟,𝑞,𝐾   𝑊,𝑞,𝑟   𝐷,𝑞,𝑟   𝐹,𝑞,𝑟

Allowed substitution hints:   𝐴(𝑟,𝑞)   𝐵(𝑟,𝑞)   + (𝑟,𝑞)   𝑆(𝑟,𝑞)   𝑇(𝑟,𝑞)   𝐿(𝑟,𝑞)   𝑀(𝑟,𝑞)   ⊥ (𝑟,𝑞)

Proof of Theorem istrnN

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trnset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		trnset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		trnset.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
4		trnset.o	. . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾)
5		trnset.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
6		trnset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
7		trnset.l	. . . 4 ⊢ 𝐿 = (Dil‘𝐾)
8		trnset.t	. . . 4 ⊢ 𝑇 = (Trn‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	trnsetN 40819	. . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑇‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})
10	9	eleq2d 2855	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))}))
11		fveq1 6881	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞))
12	11	oveq2d 7427	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑞 + (𝑓‘𝑞)) = (𝑞 + (𝐹‘𝑞)))
13	12	ineq1d 4180	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})))
14		fveq1 6881	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑟) = (𝐹‘𝑟))
15	14	oveq2d 7427	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑟 + (𝑓‘𝑟)) = (𝑟 + (𝐹‘𝑟)))
16	15	ineq1d 4180	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))
17	13, 16	eqeq12d 2785	. . . 4 ⊢ (𝑓 = 𝐹 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
18	17	2ralbidv 3235	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
19	18	elrab 3659	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))} ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))
20	10, 19	bitrdi 290	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ∩ cin 3912  {csn 4594  ‘cfv 6537  (class class class)co 7411  Atomscatm 39926  PSubSpcpsubsp 40159  +_𝑃cpadd 40458  ⊥_𝑃cpolN 40565  WAtomscwpointsN 40649  PAutcpautN 40650  DilcdilN 40765  TrnctrnN 40766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-trnN 40770

This theorem is referenced by: (None)