Theorem istendo 41423

Description: The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-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
istendo	⊢ ((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))))

Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔   𝑆,𝑓,𝑔

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

Proof of Theorem istendo

Dummy variable 𝑠 is 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	tendoset 41422	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
7	6	eleq2d 2855	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ 𝑆 ∈ {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))}))
8	3	fvexi 6896	. . . . 5 ⊢ 𝑇 ∈ V
9		fex 7225	. . . . 5 ⊢ ((𝑆:𝑇⟶𝑇 ∧ 𝑇 ∈ V) → 𝑆 ∈ V)
10	8, 9	mpan2 703	. . . 4 ⊢ (𝑆:𝑇⟶𝑇 → 𝑆 ∈ V)
11	10	3ad2ant1 1149	. . 3 ⊢ ((𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)) → 𝑆 ∈ V)
12		feq1 6684	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠:𝑇⟶𝑇 ↔ 𝑆:𝑇⟶𝑇))
13		fveq1 6881	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑠‘(𝑓 ∘ 𝑔)) = (𝑆‘(𝑓 ∘ 𝑔)))
14		fveq1 6881	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑓) = (𝑆‘𝑓))
15		fveq1 6881	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠‘𝑔) = (𝑆‘𝑔))
16	14, 15	coeq12d 5851	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))
17	13, 16	eqeq12d 2785	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))))
18	17	2ralbidv 3235	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔))))
19	14	fveq2d 6886	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑅‘(𝑠‘𝑓)) = (𝑅‘(𝑆‘𝑓)))
20	19	breq1d 5123	. . . . 5 ⊢ (𝑠 = 𝑆 → ((𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓) ↔ (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))
21	20	ralbidv 3194	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓) ↔ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))
22	12, 18, 21	3anbi123d 1462	. . 3 ⊢ (𝑠 = 𝑆 → ((𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)) ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))))
23	11, 22	elab3 3654	. 2 ⊢ (𝑆 ∈ {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))} ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓)))
24	7, 23	bitrdi 290	1 ⊢ ((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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463   class class class wbr 5113   ∘ ccom 5666  ⟶wf 6533  ‘cfv 6537  lecple 17316  LHypclh 40647  LTrncltrn 40764  trLctrl 40821  TEndoctendo 41415

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-pow 5337  ax-pr 5405  ax-un 7733

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-oprab 7415  df-mpo 7416  df-map 8825  df-tendo 41418

This theorem is referenced by:  tendotp  41424  istendod  41425  tendof  41426  tendovalco  41428