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Theorem lsmspsn 20695
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 20591 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
61, 2, 5syl2anc 585 . . 3 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
7 lsmspsn.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝑉)
83, 4lspsnsubg 20591 . . . 4 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
91, 7, 8syl2anc 585 . . 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 19517 . . 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 585 . 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 20614 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
181, 2, 17syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
1914, 15, 3, 16, 4lspsnel 20614 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
201, 7, 19syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
2118, 20anbi12d 632 . . . . . . 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 478 . . . . . 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 534 . . . . 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 3189 . . . . . . 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 3095 . . . . . 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 3189 . . . . . 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 3227 . . . . . . 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 625 . . . . . 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 298 . . . . 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 287 . . . 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 3218 . . 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 3295 . . 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 287 . 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 482 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Š ∈ LMod)
35 simprl 770 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑗 ∈ 𝐾)
362adantr 482 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑋 ∈ 𝑉)
373, 16, 14, 15, 4, 34, 35, 36lspsneli 20612 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (𝑗 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
38 simprr 772 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘˜ ∈ 𝐾)
397adantr 482 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Œ ∈ 𝑉)
403, 16, 14, 15, 4, 34, 38, 39lspsneli 20612 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (π‘˜ Β· π‘Œ) ∈ (π‘β€˜{π‘Œ}))
41 oveq1 7416 . . . . . 6 (𝑣 = (𝑗 Β· 𝑋) β†’ (𝑣 + 𝑀) = ((𝑗 Β· 𝑋) + 𝑀))
4241eqeq2d 2744 . . . . 5 (𝑣 = (𝑗 Β· 𝑋) β†’ (π‘ˆ = (𝑣 + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀)))
43 oveq2 7417 . . . . . 6 (𝑀 = (π‘˜ Β· π‘Œ) β†’ ((𝑗 Β· 𝑋) + 𝑀) = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ)))
4443eqeq2d 2744 . . . . 5 (𝑀 = (π‘˜ Β· π‘Œ) β†’ (π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
4542, 44ceqsrex2v 3647 . . . 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 585 . . 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 3218 . 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 305 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {csn 4629  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  SubGrpcsubg 19000  LSSumclsm 19502  LModclmod 20471  LSpanclspn 20582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-lsm 19504  df-mgp 19988  df-ur 20005  df-ring 20058  df-lmod 20473  df-lss 20543  df-lsp 20583
This theorem is referenced by:  lsppr  20704  baerlem3lem2  40581  baerlem5alem2  40582  baerlem5blem2  40583
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