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Theorem diclspsn 36302
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 2918 . . 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 5236 . . . . 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 2620 . . . . . . 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 36287 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))})
1211releqd 5193 . . . . 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 3679 . . . . . 6 {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊))
15 relxp 5217 . . . . . 6 Rel (𝑇 × ((TEndo‘𝐾)‘𝑊))
16 relss 5196 . . . . . 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 3198 . . . . . . 7 𝑔 ∈ V
21 vex 3198 . . . . . . 7 𝑠 ∈ V
223, 4, 5, 6, 7, 8, 9, 10, 20, 21dicopelval2 36289 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼𝑄) ↔ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
23 simprl 793 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔 = (𝑠𝐹))
24 simpll 789 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
25 simprr 795 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
26 simpl 473 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
273, 4, 5, 6lhpocnel2 35124 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
2827adantr 481 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
29 simpr 477 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
303, 4, 5, 7, 10ltrniotacl 35686 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
3126, 28, 29, 30syl3anc 1324 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
3231adantr 481 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝐹𝑇)
335, 7, 8tendocl 35874 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
3424, 25, 32, 33syl3anc 1324 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑠𝐹) ∈ 𝑇)
3523, 34eqeltrd 2699 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔𝑇)
3635, 25, 233jca 1240 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
37 simpr3 1067 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑔 = (𝑠𝐹))
38 simpr2 1066 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
3937, 38jca 554 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑔𝑇𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))) → (𝑔 = (𝑠𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)))
4036, 39impbida 876 . . . . . . . 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 2620 . . . . . . . . . . . . . 14 (Scalar‘𝑈) = (Scalar‘𝑈)
43 eqid 2620 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
445, 8, 41, 42, 43dvhbase 36191 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
4544adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
4645rexeqdv 3140 . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
48 simpr 477 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
4931adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝐹𝑇)
505, 7, 8tendoidcl 35876 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
5150ad2antrr 761 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
52 eqid 2620 . . . . . . . . . . . . . . . . . 18 ( ·𝑠𝑈) = ( ·𝑠𝑈)
535, 7, 8, 41, 52dvhopvsca 36210 . . . . . . . . . . . . . . . . 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 1326 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
5554eqeq2d 2630 . . . . . . . . . . . . . . 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 4935 . . . . . . . . . . . . . . 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 35878 . . . . . . . . . . . . . . . . . 18 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
5958adantlr 750 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
6059eqeq2d 2630 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑠 = 𝑥))
61 equcom 1943 . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 3045 . . . . . . . . . . 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 1403 . . . . . . . . 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 6177 . . . . . . . . . . . . . 14 (𝑥 = 𝑠 → (𝑥𝐹) = (𝑠𝐹))
7170eqeq2d 2630 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → (𝑔 = (𝑥𝐹) ↔ 𝑔 = (𝑠𝐹)))
7271ceqsrexv 3330 . . . . . . . . . . . 12 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)) ↔ 𝑔 = (𝑠𝐹)))
7372pm5.32i 668 . . . . . . . . . . 11 ((𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹))) ↔ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹)))
7473anbi2i 729 . . . . . . . . . 10 ((𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠𝑔 = (𝑥𝐹)))) ↔ (𝑔𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠𝐹))))
75 3anass 1040 . . . . . . . . . 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 1040 . . . . . . . . . 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 2624 . . . . . . . . . . 11 (𝑣 = ⟨𝑔, 𝑠⟩ → (𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
8180rexbidv 3048 . . . . . . . . . 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 5144 . . . . . . . . 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 2691 . . . . . . . 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 4972 . . . . . . . 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 5209 . . . 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 1323 . . 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 789 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
9145eleq2d 2685 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑥 ∈ (Base‘(Scalar‘𝑈)) ↔ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)))
9291biimpa 501 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
9350adantr 481 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
94 opelxpi 5138 . . . . . . . . . 10 ((𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9531, 93, 94syl2anc 692 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
9695adantr 481 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
975, 7, 8, 41, 52dvhvscacl 36211 . . . . . . . 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 1322 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
99 eleq1a 2694 . . . . . . 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 3027 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
102101pm4.71rd 666 . . . 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 2739 . . 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 2680 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
1055, 41, 26dvhlmod 36218 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑈 ∈ LMod)
106 eqid 2620 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
1075, 7, 8, 41, 106dvhelvbasei 36196 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
10826, 31, 93, 107syl12anc 1322 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
109 diclspsn.n . . . 4 𝑁 = (LSpan‘𝑈)
11042, 43, 106, 52, 109lspsn 18983 . . 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 692 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
112104, 111eqtr4d 2657 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 384  w3a 1036   = wceq 1481  wcel 1988  {cab 2606  wrex 2910  {crab 2913  wss 3567  {csn 4168  cop 4174   class class class wbr 4644  {copab 4703   I cid 5013   × cxp 5102  cres 5106  ccom 5108  Rel wrel 5109  cfv 5876  crio 6595  (class class class)co 6635  Basecbs 15838  Scalarcsca 15925   ·𝑠 cvsca 15926  lecple 15929  occoc 15930  LModclmod 18844  LSpanclspn 18952  Atomscatm 34369  HLchlt 34456  LHypclh 35089  LTrncltrn 35206  TEndoctendo 35859  DVecHcdvh 36186  DIsoCcdic 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-riotaBAD 34058
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-tpos 7337  df-undef 7384  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-0g 16083  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-p1 17021  df-lat 17027  df-clat 17089  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-minusg 17407  df-sbg 17408  df-mgp 18471  df-ur 18483  df-ring 18530  df-oppr 18604  df-dvdsr 18622  df-unit 18623  df-invr 18653  df-dvr 18664  df-drng 18730  df-lmod 18846  df-lss 18914  df-lsp 18953  df-lvec 19084  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457  df-llines 34603  df-lplanes 34604  df-lvols 34605  df-lines 34606  df-psubsp 34608  df-pmap 34609  df-padd 34901  df-lhyp 35093  df-laut 35094  df-ldil 35209  df-ltrn 35210  df-trl 35265  df-tendo 35862  df-edring 35864  df-dvech 36187  df-dic 36281
This theorem is referenced by:  cdlemn5pre  36308  dih1dimc  36350
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