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Theorem lsmspsn 20694
Description: Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
Hypotheses
Ref Expression
lsmspsn.v 𝑉 = (Baseβ€˜π‘Š)
lsmspsn.a + = (+gβ€˜π‘Š)
lsmspsn.f 𝐹 = (Scalarβ€˜π‘Š)
lsmspsn.k 𝐾 = (Baseβ€˜πΉ)
lsmspsn.t Β· = ( ·𝑠 β€˜π‘Š)
lsmspsn.p βŠ• = (LSSumβ€˜π‘Š)
lsmspsn.n 𝑁 = (LSpanβ€˜π‘Š)
lsmspsn.w (πœ‘ β†’ π‘Š ∈ LMod)
lsmspsn.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
lsmspsn.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
lsmspsn (πœ‘ β†’ (π‘ˆ ∈ ((π‘β€˜{𝑋}) βŠ• (π‘β€˜{π‘Œ})) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
Distinct variable groups:   𝑗,π‘˜, +   𝑗,𝐹,π‘˜   𝑗,𝐾,π‘˜   𝑗,𝑁,π‘˜   Β· ,𝑗,π‘˜   π‘ˆ,𝑗,π‘˜   𝑗,𝑉,π‘˜   𝑗,π‘Š,π‘˜   𝑗,𝑋,π‘˜   𝑗,π‘Œ,π‘˜   πœ‘,𝑗,π‘˜
Allowed substitution hints:   βŠ• (𝑗,π‘˜)
Proof of Theorem lsmspsn
Dummy variables 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmspsn.w . . . 4 (πœ‘ β†’ π‘Š ∈ LMod)
2 lsmspsn.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝑉)
3 lsmspsn.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
4 lsmspsn.n . . . . 5 𝑁 = (LSpanβ€˜π‘Š)
53, 4lspsnsubg 20590 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
61, 2, 5syl2anc 584 . . 3 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
7 lsmspsn.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝑉)
83, 4lspsnsubg 20590 . . . 4 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
91, 7, 8syl2anc 584 . . 3 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
10 lsmspsn.a . . . 4 + = (+gβ€˜π‘Š)
11 lsmspsn.p . . . 4 βŠ• = (LSSumβ€˜π‘Š)
1210, 11lsmelval 19516 . . 3 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š)) β†’ (π‘ˆ ∈ ((π‘β€˜{𝑋}) βŠ• (π‘β€˜{π‘Œ})) ↔ βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})π‘ˆ = (𝑣 + 𝑀)))
136, 9, 12syl2anc 584 . 2 (πœ‘ β†’ (π‘ˆ ∈ ((π‘β€˜{𝑋}) βŠ• (π‘β€˜{π‘Œ})) ↔ βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})π‘ˆ = (𝑣 + 𝑀)))
14 lsmspsn.f . . . . . . . . . 10 𝐹 = (Scalarβ€˜π‘Š)
15 lsmspsn.k . . . . . . . . . 10 𝐾 = (Baseβ€˜πΉ)
16 lsmspsn.t . . . . . . . . . 10 Β· = ( ·𝑠 β€˜π‘Š)
1714, 15, 3, 16, 4lspsnel 20613 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
181, 2, 17syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
1914, 15, 3, 16, 4lspsnel 20613 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
201, 7, 19syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
2118, 20anbi12d 631 . . . . . . 7 (πœ‘ β†’ ((𝑣 ∈ (π‘β€˜{𝑋}) ∧ 𝑀 ∈ (π‘β€˜{π‘Œ})) ↔ (βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ))))
2221biimpa 477 . . . . . 6 ((πœ‘ ∧ (𝑣 ∈ (π‘β€˜{𝑋}) ∧ 𝑀 ∈ (π‘β€˜{π‘Œ}))) β†’ (βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
2322biantrurd 533 . . . . 5 ((πœ‘ ∧ (𝑣 ∈ (π‘β€˜{𝑋}) ∧ 𝑀 ∈ (π‘β€˜{π‘Œ}))) β†’ (π‘ˆ = (𝑣 + 𝑀) ↔ ((βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀))))
24 r19.41v 3188 . . . . . . 7 (βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ (βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
2524rexbii 3094 . . . . . 6 (βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ βˆƒπ‘— ∈ 𝐾 (βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
26 r19.41v 3188 . . . . . 6 (βˆƒπ‘— ∈ 𝐾 (βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ (βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
27 reeanv 3226 . . . . . . 7 (βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ↔ (βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
2827anbi1i 624 . . . . . 6 ((βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 (𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ ((βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
2925, 26, 283bitrri 297 . . . . 5 (((βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋) ∧ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
3023, 29bitrdi 286 . . . 4 ((πœ‘ ∧ (𝑣 ∈ (π‘β€˜{𝑋}) ∧ 𝑀 ∈ (π‘β€˜{π‘Œ}))) β†’ (π‘ˆ = (𝑣 + 𝑀) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀))))
31302rexbidva 3217 . . 3 (πœ‘ β†’ (βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})π‘ˆ = (𝑣 + 𝑀) ↔ βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀))))
32 rexrot4 3294 . . 3 (βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 ((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)))
3331, 32bitrdi 286 . 2 (πœ‘ β†’ (βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})π‘ˆ = (𝑣 + 𝑀) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀))))
341adantr 481 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Š ∈ LMod)
35 simprl 769 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑗 ∈ 𝐾)
362adantr 481 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑋 ∈ 𝑉)
373, 16, 14, 15, 4, 34, 35, 36lspsneli 20611 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (𝑗 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
38 simprr 771 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘˜ ∈ 𝐾)
397adantr 481 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Œ ∈ 𝑉)
403, 16, 14, 15, 4, 34, 38, 39lspsneli 20611 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (π‘˜ Β· π‘Œ) ∈ (π‘β€˜{π‘Œ}))
41 oveq1 7415 . . . . . 6 (𝑣 = (𝑗 Β· 𝑋) β†’ (𝑣 + 𝑀) = ((𝑗 Β· 𝑋) + 𝑀))
4241eqeq2d 2743 . . . . 5 (𝑣 = (𝑗 Β· 𝑋) β†’ (π‘ˆ = (𝑣 + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀)))
43 oveq2 7416 . . . . . 6 (𝑀 = (π‘˜ Β· π‘Œ) β†’ ((𝑗 Β· 𝑋) + 𝑀) = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ)))
4443eqeq2d 2743 . . . . 5 (𝑀 = (π‘˜ Β· π‘Œ) β†’ (π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
4542, 44ceqsrex2v 3646 . . . 4 (((𝑗 Β· 𝑋) ∈ (π‘β€˜{𝑋}) ∧ (π‘˜ Β· π‘Œ) ∈ (π‘β€˜{π‘Œ})) β†’ (βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
4637, 40, 45syl2anc 584 . . 3 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
47462rexbidva 3217 . 2 (πœ‘ β†’ (βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 βˆƒπ‘£ ∈ (π‘β€˜{𝑋})βˆƒπ‘€ ∈ (π‘β€˜{π‘Œ})((𝑣 = (𝑗 Β· 𝑋) ∧ 𝑀 = (π‘˜ Β· π‘Œ)) ∧ π‘ˆ = (𝑣 + 𝑀)) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
4813, 33, 473bitrd 304 1 (πœ‘ β†’ (π‘ˆ ∈ ((π‘β€˜{𝑋}) βŠ• (π‘β€˜{π‘Œ})) ↔ βˆƒπ‘— ∈ 𝐾 βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {csn 4628  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  Scalarcsca 17199   ·𝑠 cvsca 17200  SubGrpcsubg 18999  LSSumclsm 19501  LModclmod 20470  LSpanclspn 20581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-lsm 19503  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lss 20542  df-lsp 20582
This theorem is referenced by:  lsppr  20703  baerlem3lem2  40576  baerlem5alem2  40577  baerlem5blem2  40578
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