Theorem istendod 41913

Description: Deduce the predicate "is 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)
istendod.1	⊢ (𝜑 → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
istendod.2	⊢ (𝜑 → 𝑆:𝑇⟶𝑇)
istendod.3	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
istendod.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))

Assertion

Ref	Expression
istendod	⊢ (𝜑 → 𝑆 ∈ 𝐸)

Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔   𝑆,𝑓,𝑔   ≤ ,𝑓   𝑅,𝑓   𝜑,𝑓,𝑔

Allowed substitution hints:   𝑅(𝑔)   𝐸(𝑓, 𝑔)   𝐻(𝑓, 𝑔)   ≤ (𝑔)   𝑉(𝑓, 𝑔)

Proof of Theorem istendod

Step	Hyp	Ref	Expression
1		istendod.2	. 2 ⊢ (𝜑 → 𝑆:𝑇⟶𝑇)
2		istendod.3	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
3	2	3expb 1138	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇)) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
4	3	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
5		istendod.4	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))
6	5	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))
7		istendod.1	. . 3 ⊢ (𝜑 → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
8		tendoset.l	. . . 4 ⊢ ≤ = (le‘𝐾)
9		tendoset.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10		tendoset.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		tendoset.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12		tendoset.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
13	8, 9, 10, 11, 12	istendo 41911	. . 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))))
14	7, 13	syl 18	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))))
15	1, 4, 6, 14	mpbir3and 1361	1 ⊢ (𝜑 → 𝑆 ∈ 𝐸)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102   ∘ ccom 5654  ⟶wf 6535  ‘cfv 6539  lecple 17441  LHypclh 41135  LTrncltrn 41252  trLctrl 41309  TEndoctendo 41903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-map 8853  df-tendo 41906

This theorem is used by:  tendoidcl  41920  tendococl  41923  tendoplcl  41932  tendo0cl  41941  tendoicl  41947  cdlemk56  42122