Theorem tendocnv 41650

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 1150	. . . . 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 41396	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇)
6		eqid 2764	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7	6, 2, 3	ltrn1o 40753	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
8	1, 5, 7	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
9		f1ococnv1 6838	. . . 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 5835	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (( I ↾ (Base‘𝐾)) ∘ ◡(𝑆‘𝐹)))
12		simp2 1151	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝑆 ∈ 𝐸)
13	6, 2, 4	tendoid 41402	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
14	1, 12, 13	syl2anc 593	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘( I ↾ (Base‘𝐾))) = ( I ↾ (Base‘𝐾)))
15	6, 2, 3	ltrn1o 40753	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
16	15	3adant2 1145	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
17		f1ococnv2 6836	. . . . . . . . 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 6873	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = (𝑆‘( I ↾ (Base‘𝐾))))
20		f1ococnv2 6836	. . . . . . . 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 2810	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = (𝑆‘(𝐹 ∘ ◡𝐹)))
23		simp3 1152	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇)
24	2, 3	ltrncnv 40775	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
25	24	3adant2 1145	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
26	2, 3, 4	tendospdi1 41649	. . . . . . 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 1393	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘(𝐹 ∘ ◡𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
28	22, 27	eqtrd 2799	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)) = ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
29	28	coeq2d 5836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹))) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹))))
30		coass 6255	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ ◡(𝑆‘𝐹)))
31		coass 6255	. . . 4 ⊢ ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)) = (◡(𝑆‘𝐹) ∘ ((𝑆‘𝐹) ∘ (𝑆‘◡𝐹)))
32	29, 30, 31	3eqtr4g 2824	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ (𝑆‘◡𝐹)))
33	10	coeq1d 5835	. . 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 41396	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ ◡𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
35	25, 34	syld3an3 1430	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹) ∈ 𝑇)
36	6, 2, 3	ltrn1o 40753	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘◡𝐹) ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
37	1, 35, 36	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
38		f1of 6808	. . . 4 ⊢ ((𝑆‘◡𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → (𝑆‘◡𝐹):(Base‘𝐾)⟶(Base‘𝐾))
39		fcoi2 6741	. . . 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 2803	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((◡(𝑆‘𝐹) ∘ (𝑆‘𝐹)) ∘ ◡(𝑆‘𝐹)) = (𝑆‘◡𝐹))
42	2, 3	ltrncnv 40775	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
43	1, 5, 42	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ 𝑇)
44	6, 2, 3	ltrn1o 40753	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡(𝑆‘𝐹) ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
45	1, 43, 44	syl2anc 593	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾))
46		f1of 6808	. . 3 ⊢ (◡(𝑆‘𝐹):(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡(𝑆‘𝐹):(Base‘𝐾)⟶(Base‘𝐾))
47		fcoi2 6741	. . 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 2808	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   I cid 5543  ^◡ccnv 5648   ↾ cres 5651   ∘ ccom 5653  ⟶wf 6519  –1-1-onto→wf1o 6522  ‘cfv 6523  Basecbs 17247  HLchlt 39979  LHypclh 40613  LTrncltrn 40730  TEndoctendo 41381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-proset 18328  df-poset 18347  df-plt 18362  df-lub 18378  df-glb 18379  df-join 18380  df-meet 18381  df-p0 18457  df-p1 18458  df-lat 18466  df-clat 18533  df-oposet 39805  df-ol 39807  df-oml 39808  df-covers 39895  df-ats 39896  df-atl 39927  df-cvlat 39951  df-hlat 39980  df-lhyp 40617  df-laut 40618  df-ldil 40733  df-ltrn 40734  df-trl 40788  df-tendo 41384

This theorem is referenced by:  tendospcanN  41652  dihjatcclem4  42050