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Theorem dih1dimatlem0 39079
Description: Lemma for dih1dimat 39081. (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 771 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = (𝑝𝐺))
2 simpl1 1193 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simprr 773 . . . . 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 37770 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
92, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
10 simpl2r 1229 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝐸)
11 simpl3 1195 . . . . . . . . 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 38855 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ≠ 𝑂))
2019simpld 498 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
212, 10, 11, 20syl3anc 1373 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐽𝑠) ∈ 𝐸)
22 simpl2l 1228 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑓𝑇)
236, 13, 14tendocl 38518 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐽𝑠) ∈ 𝐸𝑓𝑇) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
242, 21, 22, 23syl3anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
254, 5, 6, 13ltrnel 37890 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇 ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
262, 24, 9, 25syl3anc 1373 . . . . . 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 38330 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → 𝐺𝑇)
292, 9, 26, 28syl3anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺𝑇)
306, 13, 14tendocl 38518 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸𝐺𝑇) → (𝑝𝐺) ∈ 𝑇)
312, 3, 29, 30syl3anc 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) ∈ 𝑇)
321, 31eqeltrd 2838 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖𝑇)
336, 14tendococl 38523 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
342, 3, 21, 33syl3anc 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
35 simp1 1138 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3683ad2ant1 1135 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
37203adant2l 1180 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
38 simp2l 1201 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝑓𝑇)
3935, 37, 38, 23syl3anc 1373 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
4035, 39, 36, 25syl3anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
4135, 36, 40, 28syl3anc 1373 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺𝑇)
424, 5, 6, 13, 27ltrniotaval 38332 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
4335, 36, 40, 42syl3anc 1373 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
444, 5, 6, 13cdlemd 37958 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4535, 41, 39, 36, 43, 44syl311anc 1386 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺 = ((𝐽𝑠)‘𝑓))
4645adantr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4746fveq2d 6721 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
486, 13, 14tendocoval 38517 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
492, 3, 21, 22, 48syl121anc 1377 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
5047, 1, 493eqtr4d 2787 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
51 coass 6129 . . . . 5 ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠) = (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠))
5212, 6, 13, 14, 15, 16, 17, 18tendolinv 38856 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
532, 10, 11, 52syl3anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
5453coeq2d 5731 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = (𝑝 ∘ ( I ↾ 𝑇)))
556, 13, 14tendo1mulr 38522 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
562, 3, 55syl2anc 587 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
5754, 56eqtrd 2777 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = 𝑝)
5851, 57eqtr2id 2791 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
59 fveq1 6716 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑓) = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
6059eqeq2d 2748 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑖 = (𝑡𝑓) ↔ 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓)))
61 coeq1 5726 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑠) = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
6261eqeq2d 2748 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑝 = (𝑡𝑠) ↔ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠)))
6360, 62anbi12d 634 . . . . 5 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → ((𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) ↔ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))))
6463rspcev 3537 . . . 4 (((𝑝 ∘ (𝐽𝑠)) ∈ 𝐸 ∧ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6534, 50, 58, 64syl12anc 837 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6632, 3, 65jca31 518 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))))
67 simp3r 1204 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑝 = (𝑡𝑠))
6867fveq1d 6719 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝‘((𝐽𝑠)‘𝑓)) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
69 simp1l1 1268 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
70 simp2 1139 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑡𝐸)
71 simpl2r 1229 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑠𝐸)
72713ad2ant1 1135 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝐸)
736, 14tendococl 38523 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝑠𝐸) → (𝑡𝑠) ∈ 𝐸)
7469, 70, 72, 73syl3anc 1373 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑠) ∈ 𝐸)
75 simp1l3 1270 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝑂)
7669, 72, 75, 20syl3anc 1373 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐽𝑠) ∈ 𝐸)
77 simpl2l 1228 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑓𝑇)
78773ad2ant1 1135 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑓𝑇)
796, 13, 14tendocoval 38517 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑡𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
8069, 74, 76, 78, 79syl121anc 1377 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
81 coass 6129 . . . . . . . . 9 ((𝑡𝑠) ∘ (𝐽𝑠)) = (𝑡 ∘ (𝑠 ∘ (𝐽𝑠)))
8212, 6, 13, 14, 15, 16, 17, 18tendorinv 38857 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8369, 72, 75, 82syl3anc 1373 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8483coeq2d 5731 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = (𝑡 ∘ ( I ↾ 𝑇)))
856, 13, 14tendo1mulr 38522 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8669, 70, 85syl2anc 587 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8784, 86eqtrd 2777 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = 𝑡)
8881, 87syl5eq 2790 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → ((𝑡𝑠) ∘ (𝐽𝑠)) = 𝑡)
8988fveq1d 6719 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = (𝑡𝑓))
9068, 80, 893eqtr2rd 2784 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
91 simp3l 1203 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑡𝑓))
9245adantr 484 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
93923ad2ant1 1135 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝐺 = ((𝐽𝑠)‘𝑓))
9493fveq2d 6721 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
9590, 91, 943eqtr4d 2787 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑝𝐺))
9695rexlimdv3a 3205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → (∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) → 𝑖 = (𝑝𝐺)))
9796impr 458 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑖 = (𝑝𝐺))
98 simprlr 780 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑝𝐸)
9997, 98jca 515 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸))
10066, 99impbida 801 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wrex 3062   class class class wbr 5053  cmpt 5135   I cid 5454  cres 5553  ccom 5555  cfv 6380  crio 7169  Basecbs 16760  Scalarcsca 16805   ·𝑠 cvsca 16806  lecple 16809  occoc 16810  0gc0g 16944  invrcinvr 19689  LSubSpclss 19968  LSpanclspn 20008  LSAtomsclsa 36725  Atomscatm 37014  HLchlt 37101  LHypclh 37735  LTrncltrn 37852  trLctrl 37909  TEndoctendo 38503  DVecHcdvh 38829  DIsoHcdih 38979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-riotaBAD 36704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-tpos 7968  df-undef 8015  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-0g 16946  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-mgp 19505  df-ur 19517  df-ring 19564  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-drng 19769  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910  df-tendo 38506  df-edring 38508  df-dvech 38830
This theorem is referenced by:  dih1dimatlem  39080
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