Theorem tendof 41347

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

Hypotheses

Ref	Expression
tendof.h	⊢ 𝐻 = (LHyp‘𝐾)
tendof.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendof.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendof	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:𝑇⟶𝑇)

Proof of Theorem tendof

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

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
2		tendof.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendof.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		eqid 2761	. . . 4 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5		tendof.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	istendo 41344	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7		simp1 1148	. . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆‘𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)) → 𝑆:𝑇⟶𝑇)
8	6, 7	biimtrdi 255	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 → 𝑆:𝑇⟶𝑇))
9	8	imp 410	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:𝑇⟶𝑇)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5097   ∘ ccom 5647  ⟶wf 6511  ‘cfv 6515  lecple 17283  LHypclh 40568  LTrncltrn 40685  trLctrl 40742  TEndoctendo 41336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8803  df-tendo 41339

This theorem is referenced by:  tendoeq1  41348  tendocoval  41350  tendocl  41351  tendo1mul  41354  tendo1mulr  41355  tendococl  41356  tendoconid  41413  tendospass  41603  dvhlveclem  41692  dicvscacl  41775