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Theorem lsmspsn 20839
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 20735 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
61, 2, 5syl2anc 582 . . 3 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
7 lsmspsn.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝑉)
83, 4lspsnsubg 20735 . . . 4 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
91, 7, 8syl2anc 582 . . 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 19558 . . 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 582 . 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 20758 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
181, 2, 17syl2anc 582 . . . . . . . 8 (πœ‘ β†’ (𝑣 ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘— ∈ 𝐾 𝑣 = (𝑗 Β· 𝑋)))
1914, 15, 3, 16, 4lspsnel 20758 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
201, 7, 19syl2anc 582 . . . . . . . 8 (πœ‘ β†’ (𝑀 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑀 = (π‘˜ Β· π‘Œ)))
2118, 20anbi12d 629 . . . . . . 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 475 . . . . . 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 531 . . . . 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 3186 . . . . . . 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 3092 . . . . . 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 3186 . . . . . 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 3224 . . . . . . 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 622 . . . . . 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 3215 . . 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 3292 . . 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 479 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Š ∈ LMod)
35 simprl 767 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑗 ∈ 𝐾)
362adantr 479 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ 𝑋 ∈ 𝑉)
373, 16, 14, 15, 4, 34, 35, 36lspsneli 20756 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (𝑗 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
38 simprr 769 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘˜ ∈ 𝐾)
397adantr 479 . . . . 5 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ π‘Œ ∈ 𝑉)
403, 16, 14, 15, 4, 34, 38, 39lspsneli 20756 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (π‘˜ Β· π‘Œ) ∈ (π‘β€˜{π‘Œ}))
41 oveq1 7418 . . . . . 6 (𝑣 = (𝑗 Β· 𝑋) β†’ (𝑣 + 𝑀) = ((𝑗 Β· 𝑋) + 𝑀))
4241eqeq2d 2741 . . . . 5 (𝑣 = (𝑗 Β· 𝑋) β†’ (π‘ˆ = (𝑣 + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀)))
43 oveq2 7419 . . . . . 6 (𝑀 = (π‘˜ Β· π‘Œ) β†’ ((𝑗 Β· 𝑋) + 𝑀) = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ)))
4443eqeq2d 2741 . . . . 5 (𝑀 = (π‘˜ Β· π‘Œ) β†’ (π‘ˆ = ((𝑗 Β· 𝑋) + 𝑀) ↔ π‘ˆ = ((𝑗 Β· 𝑋) + (π‘˜ Β· π‘Œ))))
4542, 44ceqsrex2v 3645 . . . 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 582 . . 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 3215 . 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 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  {csn 4627  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Scalarcsca 17204   ·𝑠 cvsca 17205  SubGrpcsubg 19036  LSSumclsm 19543  LModclmod 20614  LSpanclspn 20726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-lsm 19545  df-mgp 20029  df-ur 20076  df-ring 20129  df-lmod 20616  df-lss 20687  df-lsp 20727
This theorem is referenced by:  lsppr  20848  baerlem3lem2  40884  baerlem5alem2  40885  baerlem5blem2  40886
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