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Theorem diclspsn 36800
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 2950 . . 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 relopab 5280 . . . . 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 2651 . . . . . . 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 36785 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))})
1211releqd 5237 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (Rel (𝐼𝑄) ↔ Rel {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))}))
132, 12mpbiri 248 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → Rel (𝐼𝑄))
14 ssrab2 3720 . . . . . 6 {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊))
15 relxp 5160 . . . . . 6 Rel (𝑇 × ((TEndo‘𝐾)‘𝑊))
16 relss 5240 . . . . . 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 3234 . . . . . . 7 𝑔 ∈ V
21 vex 3234 . . . . . . 7 𝑠 ∈ V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 36787 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
23 simprl 809 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔 = (𝑠𝐹))
24 simpll 805 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
25 simprr 811 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
26 simpl 472 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
273, 4, 5, 6lhpocnel2 35623 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2827adantr 480 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
29 simpr 476 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
303, 4, 5, 7, 10ltrniotacl 36184 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
3126, 28, 29, 30syl3anc 1366 . . . . . . . . . . . . 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 36372 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
3424, 25, 32, 33syl3anc 1366 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑠𝐹) ∈ 𝑇)
3523, 34eqeltrd 2730 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔𝑇)
3635, 25, 233jca 1261 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
37 simpr3 1089 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑔 = (𝑠𝐹))
38 simpr2 1088 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
3937, 38jca 553 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)))
4036, 39impbida 895 . . . . . . . 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 2651 . . . . . . . . . . . . . 14 (Scalar‘𝑈) = (Scalar‘𝑈)
43 eqid 2651 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
445, 8, 41, 42, 43dvhbase 36689 . . . . . . . . . . . . 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 3175 . . . . . . . . . . 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 805 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
48 simpr 476 . . . . . . . . . . . . . . . . 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 36374 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
5150ad2antrr 762 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
52 eqid 2651 . . . . . . . . . . . . . . . . . 18 ( ·𝑠𝑈) = ( ·𝑠𝑈)
535, 7, 8, 41, 52dvhopvsca 36708 . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
5554eqeq2d 2661 . . . . . . . . . . . . . . 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 4974 . . . . . . . . . . . . . . 15 (⟨𝑔, 𝑠⟩ = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩ ↔ (𝑔 = (𝑥𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))))
5755, 56syl6bb 276 . . . . . . . . . . . . . 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 36376 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
5958adantlr 751 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
6059eqeq2d 2661 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑠 = 𝑥))
61 equcom 1991 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑥𝑥 = 𝑠)
6260, 61syl6bb 276 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑥 = 𝑠))
6362anbi2d 740 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ((𝑔 = (𝑥𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))) ↔ (𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠)))
6457, 63bitrd 268 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠)))
65 ancom 465 . . . . . . . . . . . . 13 ((𝑔 = (𝑥𝐹) ∧ 𝑥 = 𝑠) ↔ (𝑥 = 𝑠𝑔 = (𝑥𝐹)))
6664, 65syl6bb 276 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑥 = 𝑠𝑔 = (𝑥𝐹))))
6766rexbidva 3078 . . . . . . . . . . 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 268 . . . . . . . . . 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 1445 . . . . . . . . 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 6228 . . . . . . . . . . . . . 14 (𝑥 = 𝑠 → (𝑥𝐹) = (𝑠𝐹))
7170eqeq2d 2661 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → (𝑔 = (𝑥𝐹) ↔ 𝑔 = (𝑠𝐹)))
7271ceqsrexv 3367 . . . . . . . . . . . 12 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)) ↔ 𝑔 = (𝑠𝐹)))
7372pm5.32i 670 . . . . . . . . . . 11 ((𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
7473anbi2i 730 . . . . . . . . . 10 ((𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)))) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
75 3anass 1059 . . . . . . . . . 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 1059 . . . . . . . . . 10 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
7774, 75, 763bitr4i 292 . . . . . . . . 9 ((𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
7869, 77syl6rbb 277 . . . . . . . 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 268 . . . . . . 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 2655 . . . . . . . . . . 11 (𝑣 = ⟨𝑔, 𝑠⟩ → (𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
8180rexbidv 3081 . . . . . . . . . 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 5188 . . . . . . . . 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 2722 . . . . . . . 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 opabid 5011 . . . . . . . 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 265 . . . . . . 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, 85syl6bb 276 . . . . . 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 268 . . . . 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 5253 . . . 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 1365 . . 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 805 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9145eleq2d 2716 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑥 ∈ (Base‘(Scalar‘𝑈)) ↔ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)))
9291biimpa 500 . . . . . . . 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 5182 . . . . . . . . . 10 ((𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9531, 93, 94syl2anc 694 . . . . . . . . 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 36709 . . . . . . . 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 1364 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
99 eleq1a 2725 . . . . . . 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 3060 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
102101pm4.71rd 668 . . . 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 2770 . . 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 2711 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
1055, 41, 26dvhlmod 36716 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑈 ∈ LMod)
106 eqid 2651 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
1075, 7, 8, 41, 106dvhelvbasei 36694 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
10826, 31, 93, 107syl12anc 1364 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
109 diclspsn.n . . . 4 𝑁 = (LSpan‘𝑈)
11042, 43, 106, 52, 109lspsn 19050 . . 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 694 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
112104, 111eqtr4d 2688 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 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  {crab 2945  wss 3607  {csn 4210  cop 4216   class class class wbr 4685  {copab 4745   I cid 5052   × cxp 5141  cres 5145  ccom 5147  Rel wrel 5148  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  lecple 15995  occoc 15996  LModclmod 18911  LSpanclspn 19019  Atomscatm 34868  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  TEndoctendo 36357  DVecHcdvh 36684  DIsoCcdic 36778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360  df-edring 36362  df-dvech 36685  df-dic 36779
This theorem is referenced by:  cdlemn5pre  36806  dih1dimc  36848
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