Theorem tendotp 40933

Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)

Hypotheses

Ref	Expression
tendoset.l	⊢ ≤ = (le‘𝐾)
tendoset.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoset.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
tendoset.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendotp	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))

Proof of Theorem tendotp

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tendoset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		tendoset.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendoset.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		tendoset.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
5		tendoset.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 40932	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))))
7		2fveq3 6836	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘(𝑆‘𝑓)) = (𝑅‘(𝑆‘𝐹)))
8		fveq2 6831	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑅‘𝑓) = (𝑅‘𝐹))
9	7, 8	breq12d 5108	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
10	9	rspccv 3570	. . . 4 ⊢ (∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
11	10	3ad2ant3 1135	. . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹)))
12	6, 11	biimtrdi 253	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → (𝐹 ∈ 𝑇 → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))))
13	12	3imp 1110	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095   ∘ ccom 5625  ⟶wf 6485  ‘cfv 6489  lecple 17175  LHypclh 40156  LTrncltrn 40273  trLctrl 40330  TEndoctendo 40924

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-tendo 40927

This theorem is referenced by:  tendococl  40944  tendoid  40945  tendopltp  40952  tendoicl  40968  cdlemi1  40990  tendotr  41002  cdleml1N  41148  dva1dim  41157  dialss  41218  diblss  41342