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Theorem dih1dimatlem0 37126
Description: Lemma for dih1dimat 37128. (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 778 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = (𝑝𝐺))
2 simpl1 1235 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simprr 780 . . . . 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 35817 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
92, 8syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
10 simpl2r 1292 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑠𝐸)
11 simpl3 1239 . . . . . . . . 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 36902 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ≠ 𝑂))
2019simpld 484 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
212, 10, 11, 20syl3anc 1483 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝐽𝑠) ∈ 𝐸)
22 simpl2l 1290 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑓𝑇)
236, 13, 14tendocl 36565 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐽𝑠) ∈ 𝐸𝑓𝑇) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
242, 21, 22, 23syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
254, 5, 6, 13ltrnel 35937 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇 ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊)) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
262, 24, 9, 25syl3anc 1483 . . . . . 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 36377 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → 𝐺𝑇)
292, 9, 26, 28syl3anc 1483 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺𝑇)
306, 13, 14tendocl 36565 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸𝐺𝑇) → (𝑝𝐺) ∈ 𝑇)
312, 3, 29, 30syl3anc 1483 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) ∈ 𝑇)
321, 31eqeltrd 2896 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖𝑇)
336, 14tendococl 36570 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
342, 3, 21, 33syl3anc 1483 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ (𝐽𝑠)) ∈ 𝐸)
35 simp1 1159 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3683ad2ant1 1156 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝑃𝐶 ∧ ¬ 𝑃 𝑊))
37203adant2l 1218 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐽𝑠) ∈ 𝐸)
38 simp2l 1249 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝑓𝑇)
3935, 37, 38, 23syl3anc 1483 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((𝐽𝑠)‘𝑓) ∈ 𝑇)
4035, 39, 36, 25syl3anc 1483 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊))
4135, 36, 40, 28syl3anc 1483 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺𝑇)
424, 5, 6, 13, 27ltrniotaval 36379 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ ((((𝐽𝑠)‘𝑓)‘𝑃) ∈ 𝐶 ∧ ¬ (((𝐽𝑠)‘𝑓)‘𝑃) 𝑊)) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
4335, 36, 40, 42syl3anc 1483 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃))
444, 5, 6, 13cdlemd 36005 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝐽𝑠)‘𝑓) ∈ 𝑇) ∧ (𝑃𝐶 ∧ ¬ 𝑃 𝑊) ∧ (𝐺𝑃) = (((𝐽𝑠)‘𝑓)‘𝑃)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4535, 41, 39, 36, 43, 44syl311anc 1496 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) → 𝐺 = ((𝐽𝑠)‘𝑓))
4645adantr 468 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
4746fveq2d 6421 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
486, 13, 14tendocoval 36564 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
492, 3, 21, 22, 48syl121anc 1487 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑝 ∘ (𝐽𝑠))‘𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
5047, 1, 493eqtr4d 2861 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
51 coass 5881 . . . . 5 ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠) = (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠))
5212, 6, 13, 14, 15, 16, 17, 18tendolinv 36903 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
532, 10, 11, 52syl3anc 1483 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝐽𝑠) ∘ 𝑠) = ( I ↾ 𝑇))
5453coeq2d 5499 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = (𝑝 ∘ ( I ↾ 𝑇)))
556, 13, 14tendo1mulr 36569 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑝𝐸) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
562, 3, 55syl2anc 575 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ( I ↾ 𝑇)) = 𝑝)
5754, 56eqtrd 2851 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → (𝑝 ∘ ((𝐽𝑠) ∘ 𝑠)) = 𝑝)
5851, 57syl5req 2864 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
59 fveq1 6416 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑓) = ((𝑝 ∘ (𝐽𝑠))‘𝑓))
6059eqeq2d 2827 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑖 = (𝑡𝑓) ↔ 𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓)))
61 coeq1 5494 . . . . . . 7 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑡𝑠) = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))
6261eqeq2d 2827 . . . . . 6 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → (𝑝 = (𝑡𝑠) ↔ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠)))
6360, 62anbi12d 618 . . . . 5 (𝑡 = (𝑝 ∘ (𝐽𝑠)) → ((𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) ↔ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))))
6463rspcev 3513 . . . 4 (((𝑝 ∘ (𝐽𝑠)) ∈ 𝐸 ∧ (𝑖 = ((𝑝 ∘ (𝐽𝑠))‘𝑓) ∧ 𝑝 = ((𝑝 ∘ (𝐽𝑠)) ∘ 𝑠))) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6534, 50, 58, 64syl12anc 856 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))
6632, 3, 65jca31 506 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸)) → ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))))
67 simp3r 1252 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑝 = (𝑡𝑠))
6867fveq1d 6419 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝‘((𝐽𝑠)‘𝑓)) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
69 simp1l1 1358 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
70 simp2 1160 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑡𝐸)
71 simpl2r 1292 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑠𝐸)
72713ad2ant1 1156 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝐸)
736, 14tendococl 36570 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸𝑠𝐸) → (𝑡𝑠) ∈ 𝐸)
7469, 70, 72, 73syl3anc 1483 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑠) ∈ 𝐸)
75 simp1l3 1360 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑠𝑂)
7669, 72, 75, 20syl3anc 1483 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝐽𝑠) ∈ 𝐸)
77 simpl2l 1290 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝑓𝑇)
78773ad2ant1 1156 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑓𝑇)
796, 13, 14tendocoval 36564 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑡𝑠) ∈ 𝐸 ∧ (𝐽𝑠) ∈ 𝐸) ∧ 𝑓𝑇) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
8069, 74, 76, 78, 79syl121anc 1487 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = ((𝑡𝑠)‘((𝐽𝑠)‘𝑓)))
81 coass 5881 . . . . . . . . 9 ((𝑡𝑠) ∘ (𝐽𝑠)) = (𝑡 ∘ (𝑠 ∘ (𝐽𝑠)))
8212, 6, 13, 14, 15, 16, 17, 18tendorinv 36904 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝑠𝑂) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8369, 72, 75, 82syl3anc 1483 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑠 ∘ (𝐽𝑠)) = ( I ↾ 𝑇))
8483coeq2d 5499 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = (𝑡 ∘ ( I ↾ 𝑇)))
856, 13, 14tendo1mulr 36569 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑡𝐸) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8669, 70, 85syl2anc 575 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ ( I ↾ 𝑇)) = 𝑡)
8784, 86eqtrd 2851 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡 ∘ (𝑠 ∘ (𝐽𝑠))) = 𝑡)
8881, 87syl5eq 2863 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → ((𝑡𝑠) ∘ (𝐽𝑠)) = 𝑡)
8988fveq1d 6419 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (((𝑡𝑠) ∘ (𝐽𝑠))‘𝑓) = (𝑡𝑓))
9068, 80, 893eqtr2rd 2858 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑡𝑓) = (𝑝‘((𝐽𝑠)‘𝑓)))
91 simp3l 1251 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑡𝑓))
9245adantr 468 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → 𝐺 = ((𝐽𝑠)‘𝑓))
93923ad2ant1 1156 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝐺 = ((𝐽𝑠)‘𝑓))
9493fveq2d 6421 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → (𝑝𝐺) = (𝑝‘((𝐽𝑠)‘𝑓)))
9590, 91, 943eqtr4d 2861 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) ∧ 𝑡𝐸 ∧ (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠))) → 𝑖 = (𝑝𝐺))
9695rexlimdv3a 3232 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ (𝑖𝑇𝑝𝐸)) → (∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)) → 𝑖 = (𝑝𝐺)))
9796impr 444 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑖 = (𝑝𝐺))
98 simprlr 789 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → 𝑝𝐸)
9997, 98jca 503 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓𝑇𝑠𝐸) ∧ 𝑠𝑂) ∧ ((𝑖𝑇𝑝𝐸) ∧ ∃𝑡𝐸 (𝑖 = (𝑡𝑓) ∧ 𝑝 = (𝑡𝑠)))) → (𝑖 = (𝑝𝐺) ∧ 𝑝𝐸))
10066, 99impbida 826 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wne 2989  wrex 3108   class class class wbr 4855  cmpt 4934   I cid 5231  cres 5326  ccom 5328  cfv 6110  crio 6843  Basecbs 16087  Scalarcsca 16175   ·𝑠 cvsca 16176  lecple 16179  occoc 16180  0gc0g 16324  invrcinvr 18892  LSubSpclss 19155  LSpanclspn 19197  LSAtomsclsa 34772  Atomscatm 35061  HLchlt 35148  LHypclh 35782  LTrncltrn 35899  trLctrl 35956  TEndoctendo 36550  DVecHcdvh 36876  DIsoHcdih 37026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-cnex 10286  ax-resscn 10287  ax-1cn 10288  ax-icn 10289  ax-addcl 10290  ax-addrcl 10291  ax-mulcl 10292  ax-mulrcl 10293  ax-mulcom 10294  ax-addass 10295  ax-mulass 10296  ax-distr 10297  ax-i2m1 10298  ax-1ne0 10299  ax-1rid 10300  ax-rnegex 10301  ax-rrecex 10302  ax-cnre 10303  ax-pre-lttri 10304  ax-pre-lttrn 10305  ax-pre-ltadd 10306  ax-pre-mulgt0 10307  ax-riotaBAD 34750
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-tpos 7596  df-undef 7643  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-1o 7805  df-oadd 7809  df-er 7988  df-map 8103  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-pnf 10370  df-mnf 10371  df-xr 10372  df-ltxr 10373  df-le 10374  df-sub 10562  df-neg 10563  df-nn 11315  df-2 11375  df-3 11376  df-4 11377  df-5 11378  df-6 11379  df-n0 11579  df-z 11663  df-uz 11924  df-fz 12569  df-struct 16089  df-ndx 16090  df-slot 16091  df-base 16093  df-sets 16094  df-ress 16095  df-plusg 16185  df-mulr 16186  df-sca 16188  df-vsca 16189  df-0g 16326  df-proset 17152  df-poset 17170  df-plt 17182  df-lub 17198  df-glb 17199  df-join 17200  df-meet 17201  df-p0 17263  df-p1 17264  df-lat 17270  df-clat 17332  df-mgm 17466  df-sgrp 17508  df-mnd 17519  df-grp 17649  df-minusg 17650  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 34974  df-ol 34976  df-oml 34977  df-covers 35064  df-ats 35065  df-atl 35096  df-cvlat 35120  df-hlat 35149  df-llines 35296  df-lplanes 35297  df-lvols 35298  df-lines 35299  df-psubsp 35301  df-pmap 35302  df-padd 35594  df-lhyp 35786  df-laut 35787  df-ldil 35902  df-ltrn 35903  df-trl 35957  df-tendo 36553  df-edring 36555  df-dvech 36877
This theorem is referenced by:  dih1dimatlem  37127
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