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Theorem dih1dimatlem0 36934
Description: Lemma for dih1dimat 36936. (Contributed by NM, 11-Apr-2014.)
Hypotheses
Ref Expression
dih1dimat.h 𝐻 = (LHyp‘𝐾)
dih1dimat.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dih1dimat.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dih1dimat.a 𝐴 = (LSAtoms‘𝑈)
dih1dimat.b 𝐵 = (Base‘𝐾)
dih1dimat.l = (le‘𝐾)
dih1dimat.c 𝐶 = (Atoms‘𝐾)
dih1dimat.p 𝑃 = ((oc‘𝐾)‘𝑊)
dih1dimat.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dih1dimat.r 𝑅 = ((trL‘𝐾)‘𝑊)
dih1dimat.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dih1dimat.o 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
dih1dimat.d 𝐹 = (Scalar‘𝑈)
dih1dimat.j 𝐽 = (invr𝐹)
dih1dimat.v 𝑉 = (Base‘𝑈)
dih1dimat.m · = ( ·𝑠𝑈)
dih1dimat.s 𝑆 = (LSubSp‘𝑈)
dih1dimat.n 𝑁 = (LSpan‘𝑈)
dih1dimat.z 0 = (0g𝑈)
dih1dimat.g 𝐺 = (𝑇 (𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
Assertion
Ref Expression
dih1dimatlem0 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝑖 = (𝑝𝐺) ∧ 𝑝𝐸) ↔ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))))
Distinct variable groups:   ,   𝐵,   𝑓,𝑖,𝑝,𝑠,𝑡,𝐸   𝑡,𝐹   𝐶,   𝑖,𝐺,𝑝,𝑡   𝑡,,𝐽   𝑓,𝑁,𝑠,𝑡   𝑓,,𝐾,𝑖,𝑝,𝑠,𝑡   𝑇,𝑓,,𝑖,𝑝,𝑠,𝑡   𝑈,𝑓,,𝑠,𝑡   𝑓,𝐻,,𝑖,𝑝,𝑠,𝑡   𝑓,𝑉,𝑖,𝑝,𝑠,𝑡   𝑓,𝑊,,𝑖,𝑝,𝑠,𝑡   𝑓,𝐼,𝑠   𝑖,𝑂,𝑝,𝑡   𝑃,   𝑡, ·
Allowed substitution hints:   𝐴(𝑡,𝑓,,𝑖,𝑠,𝑝)   𝐵(𝑡,𝑓,𝑖,𝑠,𝑝)   𝐶(𝑡,𝑓,𝑖,𝑠,𝑝)   𝑃(𝑡,𝑓,𝑖,𝑠,𝑝)   𝑅(𝑡,𝑓,,𝑖,𝑠,𝑝)   𝑆(𝑡,𝑓,,𝑖,𝑠,𝑝)   · (𝑓,,𝑖,𝑠,𝑝)   𝑈(𝑖,𝑝)   𝐸()   𝐹(𝑓,,𝑖,𝑠,𝑝)   𝐺(𝑓,,𝑠)   𝐼(𝑡,,𝑖,𝑝)   𝐽(𝑓,𝑖,𝑠,𝑝)   (𝑡,𝑓,𝑖,𝑠,𝑝)   𝑁(,𝑖,𝑝)   𝑂(𝑓,,𝑠)   𝑉()   0 (𝑡,𝑓,,𝑖,𝑠,𝑝)

Proof of Theorem dih1dimatlem0
StepHypRef Expression
1 simprl 809 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = (𝑝𝐺))
2 simpl1 1084 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simprr 811 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝𝐸)
4 dih1dimat.l . . . . . . . 8 = (le‘𝐾)
5 dih1dimat.c . . . . . . . 8 𝐶 = (Atoms‘𝐾)
6 dih1dimat.h . . . . . . . 8 𝐻 = (LHyp‘𝐾)
7 dih1dimat.p . . . . . . . 8 𝑃 = ((oc‘𝐾)‘𝑊)
84, 5, 6, 7lhpocnel2 35623 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
92, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
10 simpl2r 1135 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝐸)
11 simpl3 1086 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝑂)
12 dih1dimat.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
13 dih1dimat.t . . . . . . . . . . 11 𝑇 = ((LTrn‘𝐾)‘𝑊)
14 dih1dimat.e . . . . . . . . . . 11 𝐸 = ((TEndo‘𝐾)‘𝑊)
15 dih1dimat.o . . . . . . . . . . 11 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
16 dih1dimat.u . . . . . . . . . . 11 𝑈 = ((DVecH‘𝐾)‘𝑊)
17 dih1dimat.d . . . . . . . . . . 11 𝐹 = (Scalar‘𝑈)
18 dih1dimat.j . . . . . . . . . . 11 𝐽 = (invr𝐹)
1912, 6, 13, 14, 15, 16, 17, 18tendoinvcl 36710 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ≠ 𝑂))
2019simpld 474 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
212, 10, 11, 20syl3anc 1366 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐽𝑠) ∈ 𝐸)
22 simpl2l 1134 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑓𝑇)
236, 13, 14tendocl 36372 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐽𝑠) ∈ 𝐸𝑓𝑇) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
242, 21, 22, 23syl3anc 1366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
254, 5, 6, 13ltrnel 35743 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇 ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
262, 24, 9, 25syl3anc 1366 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
27 dih1dimat.g . . . . . . 7 𝐺 = (𝑇 (𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
284, 5, 6, 13, 27ltrniotacl 36184 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → 𝐺𝑇)
292, 9, 26, 28syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺𝑇)
306, 13, 14tendocl 36372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸𝐺𝑇) → (𝑝𝐺) ∈ 𝑇)
312, 3, 29, 30syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) ∈ 𝑇)
321, 31eqeltrd 2730 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖𝑇)
336, 14tendococl 36377 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
342, 3, 21, 33syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
35 simp1 1081 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3683ad2ant1 1102 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
37203adant2l 1360 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
38 simp2l 1107 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝑓𝑇)
3935, 37, 38, 23syl3anc 1366 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
4035, 39, 36, 25syl3anc 1366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
4135, 36, 40, 28syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺𝑇)
424, 5, 6, 13, 27ltrniotaval 36186 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
4335, 36, 40, 42syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
444, 5, 6, 13cdlemd 35812 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4535, 41, 39, 36, 43, 44syl311anc 1380 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺 = ((𝐽𝑠)‘𝑓))
4645adantr 480 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4746fveq2d 6233 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
486, 13, 14tendocoval 36371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
492, 3, 21, 22, 48syl121anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
5047, 1, 493eqtr4d 2695 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
51 coass 5692 . . . . 5 ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠) = (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠))
5212, 6, 13, 14, 15, 16, 17, 18tendolinv 36711 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
532, 10, 11, 52syl3anc 1366 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
5453coeq2d 5317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = (𝑝 ∘ ( I ↾ 𝑇)))
556, 13, 14tendo1mulr 36376 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
562, 3, 55syl2anc 694 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
5754, 56eqtrd 2685 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = 𝑝)
5851, 57syl5req 2698 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
59 fveq1 6228 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑓) = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
6059eqeq2d 2661 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑖 = (𝑡𝑓) ↔ 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓)))
61 coeq1 5312 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑠) = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
6261eqeq2d 2661 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑝 = (𝑡𝑠) ↔ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠)))
6360, 62anbi12d 747 . . . . 5 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → ((𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) ↔ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))))
6463rspcev 3340 . . . 4 (((𝑝 ∘ (𝐽𝑠)) ∈ 𝐸 ∧ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6534, 50, 58, 64syl12anc 1364 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6632, 3, 65jca31 556 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))))
67 simp3r 1110 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑝 = (𝑡𝑠))
6867fveq1d 6231 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝‘((𝐽𝑠)‘𝑓)) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
69 simp1l1 1174 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
70 simp2 1082 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑡𝐸)
71 simpl2r 1135 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑠𝐸)
72713ad2ant1 1102 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝐸)
736, 14tendococl 36377 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝑠𝐸) → (𝑡𝑠) ∈ 𝐸)
7469, 70, 72, 73syl3anc 1366 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑠) ∈ 𝐸)
75 simp1l3 1176 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝑂)
7669, 72, 75, 20syl3anc 1366 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐽𝑠) ∈ 𝐸)
77 simpl2l 1134 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑓𝑇)
78773ad2ant1 1102 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑓𝑇)
796, 13, 14tendocoval 36371 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑡𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
8069, 74, 76, 78, 79syl121anc 1371 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
81 coass 5692 . . . . . . . . 9 ((𝑡𝑠) ∘ (𝐽𝑠)) = (𝑡 ∘ (𝑠 ∘ (𝐽𝑠)))
8212, 6, 13, 14, 15, 16, 17, 18tendorinv 36712 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8369, 72, 75, 82syl3anc 1366 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8483coeq2d 5317 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = (𝑡 ∘ ( I ↾ 𝑇)))
856, 13, 14tendo1mulr 36376 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8669, 70, 85syl2anc 694 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8784, 86eqtrd 2685 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = 𝑡)
8881, 87syl5eq 2697 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → ((𝑡𝑠) ∘ (𝐽𝑠)) = 𝑡)
8988fveq1d 6231 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = (𝑡𝑓))
9068, 80, 893eqtr2rd 2692 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
91 simp3l 1109 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑡𝑓))
9245adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
93923ad2ant1 1102 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝐺 = ((𝐽𝑠)‘𝑓))
9493fveq2d 6233 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
9590, 91, 943eqtr4d 2695 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑝𝐺))
9695rexlimdv3a 3062 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → (∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) → 𝑖 = (𝑝𝐺)))
9796impr 648 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑖 = (𝑝𝐺))
98 simprlr 820 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑝𝐸)
9997, 98jca 553 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸))
10066, 99impbida 895 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝑖 = (𝑝𝐺) ∧ 𝑝𝐸) ↔ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942   class class class wbr 4685  cmpt 4762   I cid 5052  cres 5145  ccom 5147  cfv 5926  crio 6650  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  lecple 15995  occoc 15996  0gc0g 16147  invrcinvr 18717  LSubSpclss 18980  LSpanclspn 19019  LSAtomsclsa 34579  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763  TEndoctendo 36357  DVecHcdvh 36684  DIsoHcdih 36834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360  df-edring 36362  df-dvech 36685
This theorem is referenced by:  dih1dimatlem  36935
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