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Theorem dih1dimatlem0 37138
Description: Lemma for dih1dimat 37140. (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 811 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = (𝑝𝐺))
2 simpl1 1228 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simprr 813 . . . . 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 35827 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
92, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
10 simpl2r 1285 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝐸)
11 simpl3 1232 . . . . . . . . 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 36914 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ≠ 𝑂))
2019simpld 477 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
212, 10, 11, 20syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐽𝑠) ∈ 𝐸)
22 simpl2l 1283 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑓𝑇)
236, 13, 14tendocl 36576 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐽𝑠) ∈ 𝐸𝑓𝑇) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
242, 21, 22, 23syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
254, 5, 6, 13ltrnel 35947 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇 ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
262, 24, 9, 25syl3anc 1477 . . . . . 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 36388 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → 𝐺𝑇)
292, 9, 26, 28syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺𝑇)
306, 13, 14tendocl 36576 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸𝐺𝑇) → (𝑝𝐺) ∈ 𝑇)
312, 3, 29, 30syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) ∈ 𝑇)
321, 31eqeltrd 2840 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖𝑇)
336, 14tendococl 36581 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
342, 3, 21, 33syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
35 simp1 1131 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3683ad2ant1 1128 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
37203adant2l 1190 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
38 simp2l 1242 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝑓𝑇)
3935, 37, 38, 23syl3anc 1477 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
4035, 39, 36, 25syl3anc 1477 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
4135, 36, 40, 28syl3anc 1477 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺𝑇)
424, 5, 6, 13, 27ltrniotaval 36390 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
4335, 36, 40, 42syl3anc 1477 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
444, 5, 6, 13cdlemd 36016 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4535, 41, 39, 36, 43, 44syl311anc 1491 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺 = ((𝐽𝑠)‘𝑓))
4645adantr 472 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4746fveq2d 6358 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
486, 13, 14tendocoval 36575 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
492, 3, 21, 22, 48syl121anc 1482 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
5047, 1, 493eqtr4d 2805 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
51 coass 5816 . . . . 5 ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠) = (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠))
5212, 6, 13, 14, 15, 16, 17, 18tendolinv 36915 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
532, 10, 11, 52syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
5453coeq2d 5441 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = (𝑝 ∘ ( I ↾ 𝑇)))
556, 13, 14tendo1mulr 36580 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
562, 3, 55syl2anc 696 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
5754, 56eqtrd 2795 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = 𝑝)
5851, 57syl5req 2808 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
59 fveq1 6353 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑓) = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
6059eqeq2d 2771 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑖 = (𝑡𝑓) ↔ 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓)))
61 coeq1 5436 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑠) = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
6261eqeq2d 2771 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑝 = (𝑡𝑠) ↔ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠)))
6360, 62anbi12d 749 . . . . 5 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → ((𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) ↔ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))))
6463rspcev 3450 . . . 4 (((𝑝 ∘ (𝐽𝑠)) ∈ 𝐸 ∧ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6534, 50, 58, 64syl12anc 1475 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6632, 3, 65jca31 558 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))))
67 simp3r 1245 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑝 = (𝑡𝑠))
6867fveq1d 6356 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝‘((𝐽𝑠)‘𝑓)) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
69 simp1l1 1351 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
70 simp2 1132 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑡𝐸)
71 simpl2r 1285 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑠𝐸)
72713ad2ant1 1128 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝐸)
736, 14tendococl 36581 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝑠𝐸) → (𝑡𝑠) ∈ 𝐸)
7469, 70, 72, 73syl3anc 1477 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑠) ∈ 𝐸)
75 simp1l3 1353 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝑂)
7669, 72, 75, 20syl3anc 1477 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐽𝑠) ∈ 𝐸)
77 simpl2l 1283 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑓𝑇)
78773ad2ant1 1128 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑓𝑇)
796, 13, 14tendocoval 36575 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑡𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
8069, 74, 76, 78, 79syl121anc 1482 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
81 coass 5816 . . . . . . . . 9 ((𝑡𝑠) ∘ (𝐽𝑠)) = (𝑡 ∘ (𝑠 ∘ (𝐽𝑠)))
8212, 6, 13, 14, 15, 16, 17, 18tendorinv 36916 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8369, 72, 75, 82syl3anc 1477 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8483coeq2d 5441 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = (𝑡 ∘ ( I ↾ 𝑇)))
856, 13, 14tendo1mulr 36580 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8669, 70, 85syl2anc 696 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8784, 86eqtrd 2795 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = 𝑡)
8881, 87syl5eq 2807 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → ((𝑡𝑠) ∘ (𝐽𝑠)) = 𝑡)
8988fveq1d 6356 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = (𝑡𝑓))
9068, 80, 893eqtr2rd 2802 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
91 simp3l 1244 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑡𝑓))
9245adantr 472 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
93923ad2ant1 1128 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝐺 = ((𝐽𝑠)‘𝑓))
9493fveq2d 6358 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
9590, 91, 943eqtr4d 2805 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑝𝐺))
9695rexlimdv3a 3172 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → (∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) → 𝑖 = (𝑝𝐺)))
9796impr 650 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑖 = (𝑝𝐺))
98 simprlr 822 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑝𝐸)
9997, 98jca 555 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸))
10066, 99impbida 913 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 1072   = wceq 1632  wcel 2140  wne 2933  wrex 3052   class class class wbr 4805  cmpt 4882   I cid 5174  cres 5269  ccom 5271  cfv 6050  crio 6775  Basecbs 16080  Scalarcsca 16167   ·𝑠 cvsca 16168  lecple 16171  occoc 16172  0gc0g 16323  invrcinvr 18892  LSubSpclss 19155  LSpanclspn 19194  LSAtomsclsa 34783  Atomscatm 35072  HLchlt 35159  LHypclh 35792  LTrncltrn 35909  trLctrl 35967  TEndoctendo 36561  DVecHcdvh 36888  DIsoHcdih 37038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-riotaBAD 34761
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-tpos 7523  df-undef 7570  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-3 11293  df-4 11294  df-5 11295  df-6 11296  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-struct 16082  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-plusg 16177  df-mulr 16178  df-sca 16180  df-vsca 16181  df-0g 16325  df-preset 17150  df-poset 17168  df-plt 17180  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-p1 17262  df-lat 17268  df-clat 17330  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-grp 17647  df-minusg 17648  df-mgp 18711  df-ur 18723  df-ring 18770  df-oppr 18844  df-dvdsr 18862  df-unit 18863  df-invr 18893  df-dvr 18904  df-drng 18972  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796  df-laut 35797  df-ldil 35912  df-ltrn 35913  df-trl 35968  df-tendo 36564  df-edring 36566  df-dvech 36889
This theorem is referenced by:  dih1dimatlem  37139
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