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Theorem diclspsn 39187
Description: The value of isomorphism C is spanned by vector 𝐹. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
diclspsn.l = (le‘𝐾)
diclspsn.a 𝐴 = (Atoms‘𝐾)
diclspsn.h 𝐻 = (LHyp‘𝐾)
diclspsn.p 𝑃 = ((oc‘𝐾)‘𝑊)
diclspsn.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
diclspsn.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
diclspsn.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
diclspsn.n 𝑁 = (LSpan‘𝑈)
diclspsn.f 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)
Assertion
Ref Expression
diclspsn (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))
Distinct variable groups:   ,𝑓   𝑃,𝑓   𝐴,𝑓   𝑓,𝐻   𝑇,𝑓   𝑓,𝐾   𝑄,𝑓   𝑓,𝑊
Allowed substitution hints:   𝑈(𝑓)   𝐹(𝑓)   𝐼(𝑓)   𝑁(𝑓)

Proof of Theorem diclspsn
Dummy variables 𝑔 𝑠 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 3074 . . 3 {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {𝑣 ∣ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))}
2 relopabv 5728 . . . . 5 Rel {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))}
3 diclspsn.l . . . . . . 7 = (le‘𝐾)
4 diclspsn.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
5 diclspsn.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
6 diclspsn.p . . . . . . 7 𝑃 = ((oc‘𝐾)‘𝑊)
7 diclspsn.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 eqid 2739 . . . . . . 7 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
9 diclspsn.i . . . . . . 7 𝐼 = ((DIsoC‘𝐾)‘𝑊)
10 diclspsn.f . . . . . . 7 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)
113, 4, 5, 6, 7, 8, 9, 10dicval2 39172 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))})
1211releqd 5687 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (Rel (𝐼𝑄) ↔ Rel {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))}))
132, 12mpbiri 257 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → Rel (𝐼𝑄))
14 ssrab2 4017 . . . . . 6 {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊))
15 relxp 5606 . . . . . 6 Rel (𝑇 × ((TEndo‘𝐾)‘𝑊))
16 relss 5690 . . . . . 6 ({𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊)) → (Rel (𝑇 × ((TEndo‘𝐾)‘𝑊)) → Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
1714, 15, 16mp2 9 . . . . 5 Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}
1817a1i 11 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
19 id 22 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
20 vex 3434 . . . . . . 7 𝑔 ∈ V
21 vex 3434 . . . . . . 7 𝑠 ∈ V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 39174 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
23 simprl 767 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔 = (𝑠𝐹))
24 simpll 763 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
25 simprr 769 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
26 simpl 482 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
273, 4, 5, 6lhpocnel2 38012 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2827adantr 480 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
29 simpr 484 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
303, 4, 5, 7, 10ltrniotacl 38572 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
3126, 28, 29, 30syl3anc 1369 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
3231adantr 480 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝐹𝑇)
335, 7, 8tendocl 38760 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
3424, 25, 32, 33syl3anc 1369 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑠𝐹) ∈ 𝑇)
3523, 34eqeltrd 2840 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔𝑇)
3635, 25, 233jca 1126 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
37 simpr3 1194 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑔 = (𝑠𝐹))
38 simpr2 1193 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
3937, 38jca 511 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)))
4036, 39impbida 797 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
41 diclspsn.u . . . . . . . . . . . . . 14 𝑈 = ((DVecH‘𝐾)‘𝑊)
42 eqid 2739 . . . . . . . . . . . . . 14 (Scalar‘𝑈) = (Scalar‘𝑈)
43 eqid 2739 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
445, 8, 41, 42, 43dvhbase 39076 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
4544adantr 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
4645rexeqdv 3347 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
47 simpll 763 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
48 simpr 484 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
4931adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝐹𝑇)
505, 7, 8tendoidcl 38762 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
5150ad2antrr 722 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
52 eqid 2739 . . . . . . . . . . . . . . . . . 18 ( ·𝑠𝑈) = ( ·𝑠𝑈)
535, 7, 8, 41, 52dvhopvsca 39095 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
5447, 48, 49, 51, 53syl13anc 1370 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
5554eqeq2d 2750 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩))
5620, 21opth 5393 . . . . . . . . . . . . . . 15 (⟨𝑔, 𝑠⟩ = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩ ↔ (𝑔 = (𝑥𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))))
5755, 56bitrdi 286 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑔 = (𝑥𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇)))))
585, 7, 8tendo1mulr 38764 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
5958adantlr 711 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
6059eqeq2d 2750 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑠 = 𝑥))
61 equcom 2024 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑥𝑥 = 𝑠)
6260, 61bitrdi 286 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑥 = 𝑠))
6362anbi2d 628 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ((𝑔 = (𝑥𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))) ↔ (𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠)))
6457, 63bitrd 278 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠)))
65 ancom 460 . . . . . . . . . . . . 13 ((𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠) ↔ (𝑥 = 𝑠𝑔 = (𝑥𝐹)))
6664, 65bitrdi 286 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑥 = 𝑠𝑔 = (𝑥𝐹))))
6766rexbidva 3226 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))))
6846, 67bitrd 278 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))))
69683anbi3d 1440 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)))))
70 fveq1 6767 . . . . . . . . . . . . . 14 (𝑥 = 𝑠 → (𝑥𝐹) = (𝑠𝐹))
7170eqeq2d 2750 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → (𝑔 = (𝑥𝐹) ↔ 𝑔 = (𝑠𝐹)))
7271ceqsrexv 3586 . . . . . . . . . . . 12 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)) ↔ 𝑔 = (𝑠𝐹)))
7372pm5.32i 574 . . . . . . . . . . 11 ((𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
7473anbi2i 622 . . . . . . . . . 10 ((𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)))) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
75 3anass 1093 . . . . . . . . . 10 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)))))
76 3anass 1093 . . . . . . . . . 10 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
7774, 75, 763bitr4i 302 . . . . . . . . 9 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
7869, 77bitr2di 287 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
7940, 78bitrd 278 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
80 eqeq1 2743 . . . . . . . . . . 11 (𝑣 = ⟨𝑔, 𝑠⟩ → (𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
8180rexbidv 3227 . . . . . . . . . 10 (𝑣 = ⟨𝑔, 𝑠⟩ → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
8281rabxp 5634 . . . . . . . . 9 {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {⟨𝑔, 𝑠⟩ ∣ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))}
8382eleq2i 2831 . . . . . . . 8 (⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ↔ ⟨𝑔, 𝑠⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))})
84 opabidw 5439 . . . . . . . 8 (⟨𝑔, 𝑠⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))} ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
8583, 84bitr2i 275 . . . . . . 7 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
8679, 85bitrdi 286 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
8722, 86bitrd 278 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼𝑄) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
8887eqrelrdv2 5702 . . . 4 (((Rel (𝐼𝑄) ∧ Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}) ∧ ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐼𝑄) = {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
8913, 18, 19, 88syl21anc 834 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
90 simpll 763 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9145eleq2d 2825 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑥 ∈ (Base‘(Scalar‘𝑈)) ↔ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)))
9291biimpa 476 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
9350adantr 480 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
94 opelxpi 5625 . . . . . . . . . 10 ((𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9531, 93, 94syl2anc 583 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9695adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
975, 7, 8, 41, 52dvhvscacl 39096 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9890, 92, 96, 97syl12anc 833 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
99 eleq1a 2835 . . . . . . 7 ((𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) → (𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
10098, 99syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
101100rexlimdva 3214 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
102101pm4.71rd 562 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
103102abbidv 2808 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {𝑣 ∣ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))})
1041, 89, 1033eqtr4a 2805 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
1055, 41, 26dvhlmod 39103 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑈 ∈ LMod)
106 eqid 2739 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
1075, 7, 8, 41, 106dvhelvbasei 39081 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
10826, 31, 93, 107syl12anc 833 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
109 diclspsn.n . . . 4 𝑁 = (LSpan‘𝑈)
11042, 43, 106, 52, 109lspsn 20245 . . 3 ((𝑈 ∈ LMod ∧ ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
111105, 108, 110syl2anc 583 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
112104, 111eqtr4d 2782 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1541  wcel 2109  {cab 2716  wrex 3066  {crab 3069  wss 3891  {csn 4566  cop 4572   class class class wbr 5078  {copab 5140   I cid 5487   × cxp 5586  cres 5590  ccom 5592  Rel wrel 5593  cfv 6430  crio 7224  (class class class)co 7268  Basecbs 16893  Scalarcsca 16946   ·𝑠 cvsca 16947  lecple 16950  occoc 16951  LModclmod 20104  LSpanclspn 20214  Atomscatm 37256  HLchlt 37343  LHypclh 37977  LTrncltrn 38094  TEndoctendo 38745  DVecHcdvh 39071  DIsoCcdic 39165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-riotaBAD 36946
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-tpos 8026  df-undef 8073  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-map 8591  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-n0 12217  df-z 12303  df-uz 12565  df-fz 13222  df-struct 16829  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-mulr 16957  df-sca 16959  df-vsca 16960  df-0g 17133  df-proset 17994  df-poset 18012  df-plt 18029  df-lub 18045  df-glb 18046  df-join 18047  df-meet 18048  df-p0 18124  df-p1 18125  df-lat 18131  df-clat 18198  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-grp 18561  df-minusg 18562  df-sbg 18563  df-mgp 19702  df-ur 19719  df-ring 19766  df-oppr 19843  df-dvdsr 19864  df-unit 19865  df-invr 19895  df-dvr 19906  df-drng 19974  df-lmod 20106  df-lss 20175  df-lsp 20215  df-lvec 20346  df-oposet 37169  df-ol 37171  df-oml 37172  df-covers 37259  df-ats 37260  df-atl 37291  df-cvlat 37315  df-hlat 37344  df-llines 37491  df-lplanes 37492  df-lvols 37493  df-lines 37494  df-psubsp 37496  df-pmap 37497  df-padd 37789  df-lhyp 37981  df-laut 37982  df-ldil 38097  df-ltrn 38098  df-trl 38152  df-tendo 38748  df-edring 38750  df-dvech 39072  df-dic 39166
This theorem is referenced by:  cdlemn5pre  39193  dih1dimc  39235
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