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Theorem lbsextlem2 19925
Description: Lemma for lbsext 19929. 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 2822 . . 3 (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
2 eqidd 2822 . . 3 (𝜑 → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
3 lbsext.v . . . 4 𝑉 = (Base‘𝑊)
43a1i 11 . . 3 (𝜑𝑉 = (Base‘𝑊))
5 eqidd 2822 . . 3 (𝜑 → (+g𝑊) = (+g𝑊))
6 eqidd 2822 . . 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 19872 . . . . . . . 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})))}
1514ssrab3 4057 . . . . . . . . . . 11 𝑆 ⊆ 𝒫 𝑉
1613, 15sstrdi 3979 . . . . . . . . . 10 (𝜑𝐴 ⊆ 𝒫 𝑉)
1716sselda 3967 . . . . . . . . 9 ((𝜑𝑢𝐴) → 𝑢 ∈ 𝒫 𝑉)
1817elpwid 4553 . . . . . . . 8 ((𝜑𝑢𝐴) → 𝑢𝑉)
1918ssdifssd 4119 . . . . . . 7 ((𝜑𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ 𝑉)
20 lbsext.n . . . . . . . 8 𝑁 = (LSpan‘𝑊)
213, 20lspssv 19749 . . . . . . 7 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2212, 19, 21syl2an2r 683 . . . . . 6 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2322ralrimiva 3182 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
24 iunss 4962 . . . . 5 ( 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉 ↔ ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
2523, 24sylibr 236 . . . 4 (𝜑 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⊆ 𝑉)
269, 25eqsstrid 4015 . . 3 (𝜑𝑇𝑉)
279a1i 11 . . . 4 (𝜑𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
28 lbsext.z . . . . . 6 (𝜑𝐴 ≠ ∅)
293, 7, 20lspcl 19742 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
3012, 19, 29syl2an2r 683 . . . . . . . 8 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃)
317lssn0 19706 . . . . . . . 8 ((𝑁‘(𝑢 ∖ {𝑥})) ∈ 𝑃 → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3230, 31syl 17 . . . . . . 7 ((𝜑𝑢𝐴) → (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3332ralrimiva 3182 . . . . . 6 (𝜑 → ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
34 r19.2z 4440 . . . . . 6 ((𝐴 ≠ ∅ ∧ ∀𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅) → ∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3528, 33, 34syl2anc 586 . . . . 5 (𝜑 → ∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
36 iunn0 4982 . . . . 5 (∃𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅ ↔ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3735, 36sylib 220 . . . 4 (𝜑 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ≠ ∅)
3827, 37eqnetrd 3083 . . 3 (𝜑𝑇 ≠ ∅)
399eleq2i 2904 . . . . . . . . 9 (𝑣𝑇𝑣 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
40 eliun 4916 . . . . . . . . 9 (𝑣 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
41 difeq1 4092 . . . . . . . . . . . 12 (𝑢 = 𝑚 → (𝑢 ∖ {𝑥}) = (𝑚 ∖ {𝑥}))
4241fveq2d 6669 . . . . . . . . . . 11 (𝑢 = 𝑚 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑚 ∖ {𝑥})))
4342eleq2d 2898 . . . . . . . . . 10 (𝑢 = 𝑚 → (𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥}))))
4443cbvrexvw 3451 . . . . . . . . 9 (∃𝑢𝐴 𝑣 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
4539, 40, 443bitri 299 . . . . . . . 8 (𝑣𝑇 ↔ ∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
469eleq2i 2904 . . . . . . . . 9 (𝑤𝑇𝑤 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
47 eliun 4916 . . . . . . . . 9 (𝑤 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
48 difeq1 4092 . . . . . . . . . . . 12 (𝑢 = 𝑛 → (𝑢 ∖ {𝑥}) = (𝑛 ∖ {𝑥}))
4948fveq2d 6669 . . . . . . . . . . 11 (𝑢 = 𝑛 → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘(𝑛 ∖ {𝑥})))
5049eleq2d 2898 . . . . . . . . . 10 (𝑢 = 𝑛 → (𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
5150cbvrexvw 3451 . . . . . . . . 9 (∃𝑢𝐴 𝑤 ∈ (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
5246, 47, 513bitri 299 . . . . . . . 8 (𝑤𝑇 ↔ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
5345, 52anbi12i 628 . . . . . . 7 ((𝑣𝑇𝑤𝑇) ↔ (∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
54 reeanv 3368 . . . . . . 7 (∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) ↔ (∃𝑚𝐴 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ ∃𝑛𝐴 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
5553, 54bitr4i 280 . . . . . 6 ((𝑣𝑇𝑤𝑇) ↔ ∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))))
56 simp1l 1193 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝜑)
57 lbsext.r . . . . . . . . . . . 12 (𝜑 → [] Or 𝐴)
5856, 57syl 17 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → [] Or 𝐴)
59 simp2 1133 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝐴𝑛𝐴))
60 sorpssun 7450 . . . . . . . . . . 11 (( [] Or 𝐴 ∧ (𝑚𝐴𝑛𝐴)) → (𝑚𝑛) ∈ 𝐴)
6158, 59, 60syl2anc 586 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ∈ 𝐴)
6256, 12syl 17 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑊 ∈ LMod)
63 elssuni 4861 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ 𝐴 → (𝑚𝑛) ⊆ 𝐴)
6461, 63syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ⊆ 𝐴)
65 sspwuni 5015 . . . . . . . . . . . . . . . 16 (𝐴 ⊆ 𝒫 𝑉 𝐴𝑉)
6616, 65sylib 220 . . . . . . . . . . . . . . 15 (𝜑 𝐴𝑉)
6756, 66syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝐴𝑉)
6864, 67sstrd 3977 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚𝑛) ⊆ 𝑉)
6968ssdifssd 4119 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉)
703, 7, 20lspcl 19742 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉) → (𝑁‘((𝑚𝑛) ∖ {𝑥})) ∈ 𝑃)
7162, 69, 70syl2anc 586 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘((𝑚𝑛) ∖ {𝑥})) ∈ 𝑃)
72 simp1r 1194 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
73 ssun1 4148 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
74 ssdif 4116 . . . . . . . . . . . . . 14 (𝑚 ⊆ (𝑚𝑛) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
7573, 74mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
763, 20lspss 19750 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑚 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥})) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
7762, 69, 75, 76syl3anc 1367 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑚 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
78 simp3l 1197 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})))
7977, 78sseldd 3968 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑣 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
80 ssun2 4149 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
81 ssdif 4116 . . . . . . . . . . . . . 14 (𝑛 ⊆ (𝑚𝑛) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
8280, 81mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥}))
833, 20lspss 19750 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ ((𝑚𝑛) ∖ {𝑥}) ⊆ 𝑉 ∧ (𝑛 ∖ {𝑥}) ⊆ ((𝑚𝑛) ∖ {𝑥})) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
8462, 69, 82, 83syl3anc 1367 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → (𝑁‘(𝑛 ∖ {𝑥})) ⊆ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
85 simp3r 1198 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))
8684, 85sseldd 3968 . . . . . . . . . . 11 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → 𝑤 ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
87 eqid 2821 . . . . . . . . . . . 12 (Scalar‘𝑊) = (Scalar‘𝑊)
88 eqid 2821 . . . . . . . . . . . 12 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
89 eqid 2821 . . . . . . . . . . . 12 (+g𝑊) = (+g𝑊)
90 eqid 2821 . . . . . . . . . . . 12 ( ·𝑠𝑊) = ( ·𝑠𝑊)
9187, 88, 89, 90, 7lsscl 19708 . . . . . . . . . . 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})))
9271, 72, 79, 86, 91syl13anc 1368 . . . . . . . . . 10 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥})))
93 difeq1 4092 . . . . . . . . . . . 12 (𝑢 = (𝑚𝑛) → (𝑢 ∖ {𝑥}) = ((𝑚𝑛) ∖ {𝑥}))
9493fveq2d 6669 . . . . . . . . . . 11 (𝑢 = (𝑚𝑛) → (𝑁‘(𝑢 ∖ {𝑥})) = (𝑁‘((𝑚𝑛) ∖ {𝑥})))
9594eliuni 4918 . . . . . . . . . 10 (((𝑚𝑛) ∈ 𝐴 ∧ ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ (𝑁‘((𝑚𝑛) ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
9661, 92, 95syl2anc 586 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
9796, 9eleqtrrdi 2924 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴) ∧ (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥})))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇)
98973expia 1117 . . . . . . 7 (((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑚𝐴𝑛𝐴)) → ((𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
9998rexlimdvva 3294 . . . . . 6 ((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) → (∃𝑚𝐴𝑛𝐴 (𝑣 ∈ (𝑁‘(𝑚 ∖ {𝑥})) ∧ 𝑤 ∈ (𝑁‘(𝑛 ∖ {𝑥}))) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
10055, 99syl5bi 244 . . . . 5 ((𝜑𝑟 ∈ (Base‘(Scalar‘𝑊))) → ((𝑣𝑇𝑤𝑇) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))
101100exp4b 433 . . . 4 (𝜑 → (𝑟 ∈ (Base‘(Scalar‘𝑊)) → (𝑣𝑇 → (𝑤𝑇 → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇))))
1021013imp2 1345 . . 3 ((𝜑 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑣𝑇𝑤𝑇)) → ((𝑟( ·𝑠𝑊)𝑣)(+g𝑊)𝑤) ∈ 𝑇)
1031, 2, 4, 5, 6, 8, 26, 38, 102islssd 19701 . 2 (𝜑𝑇𝑃)
104 eldifi 4103 . . . . . . 7 (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → 𝑦 𝐴)
105104adantl 484 . . . . . 6 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → 𝑦 𝐴)
106 eldifn 4104 . . . . . . . . . 10 (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → ¬ 𝑦 ∈ {𝑥})
107106ad2antlr 725 . . . . . . . . 9 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → ¬ 𝑦 ∈ {𝑥})
108 eldif 3946 . . . . . . . . . 10 (𝑦 ∈ (𝑢 ∖ {𝑥}) ↔ (𝑦𝑢 ∧ ¬ 𝑦 ∈ {𝑥}))
1093, 20lspssid 19751 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ (𝑢 ∖ {𝑥}) ⊆ 𝑉) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
11012, 19, 109syl2an2r 683 . . . . . . . . . . . 12 ((𝜑𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
111110adantlr 713 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑢 ∖ {𝑥}) ⊆ (𝑁‘(𝑢 ∖ {𝑥})))
112111sseld 3966 . . . . . . . . . 10 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑦 ∈ (𝑢 ∖ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
113108, 112syl5bir 245 . . . . . . . . 9 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → ((𝑦𝑢 ∧ ¬ 𝑦 ∈ {𝑥}) → 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
114107, 113mpan2d 692 . . . . . . . 8 (((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) ∧ 𝑢𝐴) → (𝑦𝑢𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
115114reximdva 3274 . . . . . . 7 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → (∃𝑢𝐴 𝑦𝑢 → ∃𝑢𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥}))))
116 eluni2 4836 . . . . . . 7 (𝑦 𝐴 ↔ ∃𝑢𝐴 𝑦𝑢)
117 eliun 4916 . . . . . . 7 (𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ↔ ∃𝑢𝐴 𝑦 ∈ (𝑁‘(𝑢 ∖ {𝑥})))
118115, 116, 1173imtr4g 298 . . . . . 6 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → (𝑦 𝐴𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
119105, 118mpd 15 . . . . 5 ((𝜑𝑦 ∈ ( 𝐴 ∖ {𝑥})) → 𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
120119ex 415 . . . 4 (𝜑 → (𝑦 ∈ ( 𝐴 ∖ {𝑥}) → 𝑦 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))))
121120ssrdv 3973 . . 3 (𝜑 → ( 𝐴 ∖ {𝑥}) ⊆ 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥})))
122121, 9sseqtrrdi 4018 . 2 (𝜑 → ( 𝐴 ∖ {𝑥}) ⊆ 𝑇)
123103, 122jca 514 1 (𝜑 → (𝑇𝑃 ∧ ( 𝐴 ∖ {𝑥}) ⊆ 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139  {crab 3142  cdif 3933  cun 3934  wss 3936  c0 4291  𝒫 cpw 4539  {csn 4561   cuni 4832   ciun 4912   Or wor 5468  cfv 6350  (class class class)co 7150   [] crpss 7442  Basecbs 16477  +gcplusg 16559  Scalarcsca 16562   ·𝑠 cvsca 16563  LModclmod 19628  LSubSpclss 19697  LSpanclspn 19737  LBasisclbs 19840  LVecclvec 19868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rpss 7443  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-plusg 16572  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-mgp 19234  df-ur 19246  df-ring 19293  df-lmod 19630  df-lss 19698  df-lsp 19738  df-lvec 19869
This theorem is referenced by:  lbsextlem3  19926
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