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Theorem lbsextlem2 19078
Description: Lemma for lbsext 19082. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (Contributed by Mario Carneiro, 25-Jun-2014.)
Hypotheses
Ref Expression
lbsext.v 𝑉 = (Base‘𝑊)
lbsext.j 𝐽 = (LBasis‘𝑊)
lbsext.n 𝑁 = (LSpan‘𝑊)
lbsext.w (𝜑𝑊 ∈ LVec)
lbsext.c (𝜑𝐶𝑉)
lbsext.x (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))
lbsext.s 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
lbsext.p 𝑃 = (LSubSp‘𝑊)
lbsext.a (𝜑𝐴𝑆)
lbsext.z (𝜑𝐴 ≠ ∅)
lbsext.r (𝜑 → [] Or 𝐴)
lbsext.t 𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))
Assertion
Ref Expression
lbsextlem2 (𝜑 → (𝑇𝑃 ∧ ( 𝐴 ∖ {𝑥}) ⊆ 𝑇))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑢,𝜑   𝑢,𝑆,𝑥   𝑥,𝑧,𝐶   𝑧,𝑢,𝑁,𝑥   𝑢,𝑉,𝑥,𝑧   𝑢,𝑊,𝑥   𝑢,𝐴,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝐶(𝑢)   𝑃(𝑥,𝑧,𝑢)   𝑆(𝑧)   𝑇(𝑥,𝑧,𝑢)   𝐽(𝑧,𝑢)   𝑊(𝑧)

Proof of Theorem lbsextlem2
Dummy variables 𝑚 𝑛 𝑟 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2622 . . 3 (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2 eqidd 2622 . . 3 (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
3 lbsext.v . . . 4 𝑉 = (Base‘𝑊)
43a1i 11 . . 3 (𝜑𝑉 = (Base‘𝑊))
5 eqidd 2622 . . 3 (𝜑 → (+g𝑊) = (+g𝑊))
6 eqidd 2622 . . 3 (𝜑 → ( ·𝑠𝑊) = ( ·𝑠𝑊))
7 lbsext.p . . . 4 𝑃 = (LSubSp‘𝑊)
87a1i 11 . . 3 (𝜑𝑃 = (LSubSp‘𝑊))
9 lbsext.t . . . 4 𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))
10 lbsext.w . . . . . . . 8 (𝜑𝑊 ∈ LVec)
11 lveclmod 19025 . . . . . . . 8 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
1210, 11syl 17 . . . . . . 7 (𝜑𝑊 ∈ LMod)
13 lbsext.a . . . . . . . . . . 11 (𝜑𝐴𝑆)
14 lbsext.s . . . . . . . . . . . 12 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}
15 ssrab2 3666 . . . . . . . . . . . 12 {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} ⊆ 𝒫 𝑉
1614, 15eqsstri 3614 . . . . . . . . . . 11 𝑆 ⊆ 𝒫 𝑉
1713, 16syl6ss 3595 . . . . . . . . . 10 (𝜑𝐴 ⊆ 𝒫 𝑉)
1817sselda 3583 . . . . . . . . 9 ((𝜑𝑢𝐴) → 𝑢 ∈ 𝒫 𝑉)
1918elpwid 4141 . . . . . . . 8 ((𝜑𝑢𝐴) → 𝑢𝑉)
2019ssdifssd 3726 . . . . . . 7 ((𝜑𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
21 lbsext.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
223, 21lspssv 18902 . . . . . . 7 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2312, 20, 22syl2an2r 875 . . . . . 6 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2423ralrimiva 2960 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
25 iunss 4527 . . . . 5 ( 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉 ↔ ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2624, 25sylibr 224 . . . 4 (𝜑 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
279, 26syl5eqss 3628 . . 3 (𝜑𝑇𝑉)
289a1i 11 . . . 4 (𝜑𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
29 lbsext.z . . . . . 6 (𝜑𝐴 ≠ ∅)
303, 7, 21lspcl 18895 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
3112, 20, 30syl2an2r 875 . . . . . . . 8 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
327lssn0 18860 . . . . . . . 8 ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3331, 32syl 17 . . . . . . 7 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3433ralrimiva 2960 . . . . . 6 (𝜑 → ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
35 r19.2z 4032 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3629, 34, 35syl2anc 692 . . . . 5 (𝜑 → ∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
37 iunn0 4546 . . . . 5 (∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅ ↔ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3836, 37sylib 208 . . . 4 (𝜑 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3928, 38eqnetrd 2857 . . 3 (𝜑𝑇 ≠ ∅)
409eleq2i 2690 . . . . . . . . 9 (𝑣𝑇𝑣 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
41 eliun 4490 . . . . . . . . 9 (𝑣 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
42 difeq1 3699 . . . . . . . . . . . 12 (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
4342fveq2d 6152 . . . . . . . . . . 11 (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
4443eleq2d 2684 . . . . . . . . . 10 (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
4544cbvrexv 3160 . . . . . . . . 9 (∃𝑢𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
4640, 41, 453bitri 286 . . . . . . . 8 (𝑣𝑇 ↔ ∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
479eleq2i 2690 . . . . . . . . 9 (𝑤𝑇𝑤 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
48 eliun 4490 . . . . . . . . 9 (𝑤 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
49 difeq1 3699 . . . . . . . . . . . 12 (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
5049fveq2d 6152 . . . . . . . . . . 11 (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
5150eleq2d 2684 . . . . . . . . . 10 (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
5251cbvrexv 3160 . . . . . . . . 9 (∃𝑢𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
5347, 48, 523bitri 286 . . . . . . . 8 (𝑤𝑇 ↔ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
5446, 53anbi12i 732 . . . . . . 7 ((𝑣𝑇𝑤𝑇) ↔ (∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
55 reeanv 3097 . . . . . . 7 (∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) ↔ (∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
5654, 55bitr4i 267 . . . . . 6 ((𝑣𝑇𝑤𝑇) ↔ ∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
57 simp1l 1083 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝜑)
58 lbsext.r . . . . . . . . . . . 12 (𝜑 → [] Or 𝐴)
5957, 58syl 17 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → [] Or 𝐴)
60 simp2 1060 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝐴𝑛𝐴))
61 sorpssun 6897 . . . . . . . . . . 11 (( [] Or 𝐴 ∧ (𝑚𝐴𝑛𝐴)) → (𝑚𝑛) ∈ 𝐴)
6259, 60, 61syl2anc 692 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ∈ 𝐴)
6357, 12syl 17 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑊 ∈ LMod)
64 elssuni 4433 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ 𝐴 → (𝑚𝑛) ⊆ 𝐴)
6562, 64syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ⊆ 𝐴)
66 sspwuni 4577 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ 𝒫 𝑉 𝐴𝑉)
6717, 66sylib 208 . . . . . . . . . . . . . . 15 (𝜑 𝐴𝑉)
6857, 67syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝐴𝑉)
6965, 68sstrd 3593 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ⊆ 𝑉)
7069ssdifssd 3726 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉)
713, 7, 21lspcl 18895 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚𝑛) ∖ {𝑥})) ∈ 𝑃)
7263, 70, 71syl2anc 692 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘((𝑚𝑛) ∖ {𝑥})) ∈ 𝑃)
73 simp1r 1084 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
74 ssun1 3754 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
75 ssdif 3723 . . . . . . . . . . . . . 14 (𝑚 ⊆ (𝑚𝑛) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
7674, 75mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
773, 21lspss 18903 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥})) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
7863, 70, 76, 77syl3anc 1323 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
79 simp3l 1087 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
8078, 79sseldd 3584 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
81 ssun2 3755 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
82 ssdif 3723 . . . . . . . . . . . . . 14 (𝑛 ⊆ (𝑚𝑛) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
8381, 82mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
843, 21lspss 18903 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥})) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
8563, 70, 83, 84syl3anc 1323 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
86 simp3r 1088 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
8785, 86sseldd 3584 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
88 eqid 2621 . . . . . . . . . . . 12 (Scalar‘𝑊) = (Scalar‘𝑊)
89 eqid 2621 . . . . . . . . . . . 12 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
90 eqid 2621 . . . . . . . . . . . 12 (+g𝑊) = (+g𝑊)
91 eqid 2621 . . . . . . . . . . . 12 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9288, 89, 90, 91, 7lsscl 18862 . . . . . . . . . . 11 (((𝑁‘((𝑚𝑛) ∖ {𝑥})) ∈ 𝑃 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑣 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
9372, 73, 80, 87, 92syl13anc 1325 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
94 difeq1 3699 . . . . . . . . . . . 12 (𝑢 = (𝑚𝑛) → (𝑢 ∖ {𝑥}) = ((𝑚𝑛) ∖ {𝑥}))
9594fveq2d 6152 . . . . . . . . . . 11 (𝑢 = (𝑚𝑛) → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘((𝑚𝑛) ∖ {𝑥})))
9695eliuni 4492 . . . . . . . . . 10 (((𝑚𝑛) ∈ 𝐴 ∧ ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
9762, 93, 96syl2anc 692 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
9897, 9syl6eleqr 2709 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇)
99983expia 1264 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴)) → ((𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
10099rexlimdvva 3031 . . . . . 6 ((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) → (∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
10156, 100syl5bi 232 . . . . 5 ((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) → ((𝑣𝑇𝑤𝑇) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
102101exp4b 631 . . . 4 (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣𝑇 → (𝑤𝑇 → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))))
1031023imp2 1279 . . 3 ((𝜑 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑣𝑇𝑤𝑇)) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇)
1041, 2, 4, 5, 6, 8, 27, 39, 103islssd 18855 . 2 (𝜑𝑇𝑃)
105 eldifi 3710 . . . . . . 7 (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → 𝑦 𝐴)
106105adantl 482 . . . . . 6 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → 𝑦 𝐴)
107 eldifn 3711 . . . . . . . . . 10 (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
108107ad2antlr 762 . . . . . . . . 9 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → ¬ 𝑦 ∈ {𝑥})
109 eldif 3565 . . . . . . . . . 10 (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
1103, 21lspssid 18904 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
11112, 20, 110syl2an2r 875 . . . . . . . . . . . 12 ((𝜑𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112111adantlr 750 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
113112sseld 3582 . . . . . . . . . 10 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑦 ∈ (𝑢 ∖ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
114109, 113syl5bir 233 . . . . . . . . 9 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → ((𝑦𝑢 ∧ ¬ 𝑦 ∈ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115108, 114mpan2d 709 . . . . . . . 8 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑦𝑢𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116115reximdva 3011 . . . . . . 7 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → (∃𝑢𝐴 𝑦𝑢 → ∃𝑢𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
117 eluni2 4406 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑢𝐴 𝑦𝑢)
118 eliun 4490 . . . . . . 7 (𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
119116, 117, 1183imtr4g 285 . . . . . 6 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → (𝑦 𝐴𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
120106, 119mpd 15 . . . . 5 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → 𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
121120ex 450 . . . 4 (𝜑 → (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → 𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
122121ssrdv 3589 . . 3 (𝜑 → ( 𝐴 ∖ {𝑥}) ⊆ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
123122, 9syl6sseqr 3631 . 2 (𝜑 → ( 𝐴 ∖ {𝑥}) ⊆ 𝑇)
124104, 123jca 554 1 (𝜑 → (𝑇𝑃 ∧ ( 𝐴 ∖ {𝑥}) ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3552  cun 3553  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402   ciun 4485   Or wor 4994  cfv 5847  (class class class)co 6604   [] crpss 6889  Basecbs 15781  +gcplusg 15862  Scalarcsca 15865   ·𝑠 cvsca 15866  LModclmod 18784  LSubSpclss 18851  LSpanclspn 18890  LBasisclbs 18993  LVecclvec 19021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-rpss 6890  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lvec 19022
This theorem is referenced by:  lbsextlem3  19079
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