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Theorem dih1dimatlem0 41027
Description: Lemma for dih1dimat 41029. (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 769 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = (𝑝𝐺))
2 simpl1 1188 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simprr 771 . . . . 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 39718 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
92, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
10 simpl2r 1224 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝐸)
11 simpl3 1190 . . . . . . . . 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 40803 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ≠ 𝑂))
2019simpld 493 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
212, 10, 11, 20syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐽𝑠) ∈ 𝐸)
22 simpl2l 1223 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑓𝑇)
236, 13, 14tendocl 40466 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐽𝑠) ∈ 𝐸𝑓𝑇) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
242, 21, 22, 23syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
254, 5, 6, 13ltrnel 39838 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇 ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
262, 24, 9, 25syl3anc 1368 . . . . . 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 40278 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → 𝐺𝑇)
292, 9, 26, 28syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺𝑇)
306, 13, 14tendocl 40466 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸𝐺𝑇) → (𝑝𝐺) ∈ 𝑇)
312, 3, 29, 30syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) ∈ 𝑇)
321, 31eqeltrd 2826 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖𝑇)
336, 14tendococl 40471 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
342, 3, 21, 33syl3anc 1368 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
35 simp1 1133 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3683ad2ant1 1130 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
37203adant2l 1175 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
38 simp2l 1196 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝑓𝑇)
3935, 37, 38, 23syl3anc 1368 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
4035, 39, 36, 25syl3anc 1368 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
4135, 36, 40, 28syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺𝑇)
424, 5, 6, 13, 27ltrniotaval 40280 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
4335, 36, 40, 42syl3anc 1368 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
444, 5, 6, 13cdlemd 39906 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4535, 41, 39, 36, 43, 44syl311anc 1381 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺 = ((𝐽𝑠)‘𝑓))
4645adantr 479 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4746fveq2d 6905 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
486, 13, 14tendocoval 40465 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
492, 3, 21, 22, 48syl121anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
5047, 1, 493eqtr4d 2776 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
51 coass 6276 . . . . 5 ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠) = (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠))
5212, 6, 13, 14, 15, 16, 17, 18tendolinv 40804 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
532, 10, 11, 52syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
5453coeq2d 5869 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = (𝑝 ∘ ( I ↾ 𝑇)))
556, 13, 14tendo1mulr 40470 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
562, 3, 55syl2anc 582 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
5754, 56eqtrd 2766 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = 𝑝)
5851, 57eqtr2id 2779 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
59 fveq1 6900 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑓) = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
6059eqeq2d 2737 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑖 = (𝑡𝑓) ↔ 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓)))
61 coeq1 5864 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑠) = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
6261eqeq2d 2737 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑝 = (𝑡𝑠) ↔ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠)))
6360, 62anbi12d 630 . . . . 5 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → ((𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) ↔ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))))
6463rspcev 3608 . . . 4 (((𝑝 ∘ (𝐽𝑠)) ∈ 𝐸 ∧ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6534, 50, 58, 64syl12anc 835 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6632, 3, 65jca31 513 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))))
67 simp3r 1199 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑝 = (𝑡𝑠))
6867fveq1d 6903 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝‘((𝐽𝑠)‘𝑓)) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
69 simp1l1 1263 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
70 simp2 1134 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑡𝐸)
71 simpl2r 1224 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑠𝐸)
72713ad2ant1 1130 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝐸)
736, 14tendococl 40471 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝑠𝐸) → (𝑡𝑠) ∈ 𝐸)
7469, 70, 72, 73syl3anc 1368 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑠) ∈ 𝐸)
75 simp1l3 1265 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝑂)
7669, 72, 75, 20syl3anc 1368 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐽𝑠) ∈ 𝐸)
77 simpl2l 1223 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑓𝑇)
78773ad2ant1 1130 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑓𝑇)
796, 13, 14tendocoval 40465 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑡𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
8069, 74, 76, 78, 79syl121anc 1372 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
81 coass 6276 . . . . . . . . 9 ((𝑡𝑠) ∘ (𝐽𝑠)) = (𝑡 ∘ (𝑠 ∘ (𝐽𝑠)))
8212, 6, 13, 14, 15, 16, 17, 18tendorinv 40805 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8369, 72, 75, 82syl3anc 1368 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8483coeq2d 5869 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = (𝑡 ∘ ( I ↾ 𝑇)))
856, 13, 14tendo1mulr 40470 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8669, 70, 85syl2anc 582 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8784, 86eqtrd 2766 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = 𝑡)
8881, 87eqtrid 2778 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → ((𝑡𝑠) ∘ (𝐽𝑠)) = 𝑡)
8988fveq1d 6903 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = (𝑡𝑓))
9068, 80, 893eqtr2rd 2773 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
91 simp3l 1198 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑡𝑓))
9245adantr 479 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
93923ad2ant1 1130 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝐺 = ((𝐽𝑠)‘𝑓))
9493fveq2d 6905 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
9590, 91, 943eqtr4d 2776 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑝𝐺))
9695rexlimdv3a 3149 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → (∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) → 𝑖 = (𝑝𝐺)))
9796impr 453 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑖 = (𝑝𝐺))
98 simprlr 778 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑝𝐸)
9997, 98jca 510 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸))
10066, 99impbida 799 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wrex 3060   class class class wbr 5153  cmpt 5236   I cid 5579  cres 5684  ccom 5686  cfv 6554  crio 7379  Basecbs 17213  Scalarcsca 17269   ·𝑠 cvsca 17270  lecple 17273  occoc 17274  0gc0g 17454  invrcinvr 20369  LSubSpclss 20908  LSpanclspn 20948  LSAtomsclsa 38672  Atomscatm 38961  HLchlt 39048  LHypclh 39683  LTrncltrn 39800  trLctrl 39857  TEndoctendo 40451  DVecHcdvh 40777  DIsoHcdih 40927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-riotaBAD 38651
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-undef 8288  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-sca 17282  df-vsca 17283  df-0g 17456  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-oppr 20316  df-dvdsr 20339  df-unit 20340  df-invr 20370  df-dvr 20383  df-drng 20709  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199  df-lines 39200  df-psubsp 39202  df-pmap 39203  df-padd 39495  df-lhyp 39687  df-laut 39688  df-ldil 39803  df-ltrn 39804  df-trl 39858  df-tendo 40454  df-edring 40456  df-dvech 40778
This theorem is referenced by:  dih1dimatlem  41028
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