Theorem istendod 41393

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 1136	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇)) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
4	3	ralrimivva 3208	. 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 3157	. 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 41391	. . 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 1359	1 ⊢ (𝜑 → 𝑆 ∈ 𝐸)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5104   ∘ ccom 5655  ⟶wf 6521  ‘cfv 6525  lecple 17305  LHypclh 40615  LTrncltrn 40732  trLctrl 40789  TEndoctendo 41383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-tendo 41386

This theorem is referenced by:  tendoidcl  41400  tendococl  41403  tendoplcl  41412  tendo0cl  41421  tendoicl  41427  cdlemk56  41602