Theorem tendocnv 38317

Description: Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)

Hypotheses

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

Assertion

Ref	Expression
tendocnv	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹))

Proof of Theorem tendocnv

Step	Hyp	Ref	Expression
1		simp1 1133	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		tendosp.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		tendosp.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
4		tendosp.e	. . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
5	2, 3, 4	tendocl 38063	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇)
6		eqid 2798	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7	6, 2, 3	ltrn1o 37420	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8	1, 5, 7	syl2anc 587	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		f1ococnv1 6618	. . . 4 ⊢ ((𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
10	8, 9	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
11	10	coeq1d 5696	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)))
12		simp2 1134	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝑆 ∈ 𝐸)
13	6, 2, 4	tendoid 38069	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
14	1, 12, 13	syl2anc 587	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
15	6, 2, 3	ltrn1o 37420	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
16	15	3adant2 1128	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
17		f1ococnv2 6616	. . . . . . . . 9 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝐾)))
18	16, 17	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝐹 ∘ ◡𝐹) = ( I ↾ (Base‘𝐾)))
19	18	fveq2d 6649	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = (𝑆‘( I ↾ (Base‘𝐾))))
20		f1ococnv2 6616	. . . . . . . 8 ⊢ ((𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
21	8, 20	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ( I ↾ (Base‘𝐾)))
22	14, 19, 21	3eqtr4rd 2844	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = (𝑆‘(𝐹 ∘ ◡𝐹)))
23		simp3 1135	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
24	2, 3	ltrncnv 37442	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
25	24	3adant2 1128	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	2, 3, 4	tendospdi1 38316	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ ◡𝐹 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ ◡𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
27	1, 12, 23, 25, 26	syl13anc 1369	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
28	22, 27	eqtrd 2833	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
29	28	coeq2d 5697	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹))) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹))))
30		coass 6085	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)))
31		coass 6085	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
32	29, 30, 31	3eqtr4g 2858	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)))
33	10	coeq1d 5696	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)) = (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)))
34	2, 3, 4	tendocl 38063	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ ◡𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
35	25, 34	syld3an3 1406	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
36	6, 2, 3	ltrn1o 37420	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘◡𝐹) ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
37	1, 35, 36	syl2anc 587	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
38		f1of 6590	. . . 4 ⊢ ((𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (𝑆‘◡𝐹):(Base‘𝐾)⟶(Base‘𝐾))
39		fcoi2 6527	. . . 4 ⊢ ((𝑆‘◡𝐹):(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)) = (𝑆‘◡𝐹))
40	37, 38, 39	3syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ (𝑆‘◡𝐹)) = (𝑆‘◡𝐹))
41	32, 33, 40	3eqtrd 2837	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (𝑆‘◡𝐹))
42	2, 3	ltrncnv 37442	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
43	1, 5, 42	syl2anc 587	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
44	6, 2, 3	ltrn1o 37420	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡(𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 43, 44	syl2anc 587	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6590	. . 3 ⊢ (◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡(𝑆‘𝐹):(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi2 6527	. . 3 ⊢ (◡(𝑆‘𝐹):(Base‘𝐾)⟶(Base‘𝐾) → (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)) = ◡(𝑆‘𝐹))
48	45, 46, 47	3syl 18	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)) = ◡(𝑆‘𝐹))
49	11, 41, 48	3eqtr3rd 2842	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   I cid 5424  ^◡ccnv 5518   ↾ cres 5521   ∘ ccom 5523  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  Basecbs 16475  HLchlt 36646  LHypclh 37280  LTrncltrn 37397  TEndoctendo 38048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-lhyp 37284  df-laut 37285  df-ldil 37400  df-ltrn 37401  df-trl 37455  df-tendo 38051

This theorem is referenced by:  tendospcanN  38319  dihjatcclem4  38717