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Theorem lsmspsn 20560
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 20456 . . . 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 20456 . . . 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 19436 . . 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 20479 . . . . . . . . 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 20479 . . . . . . . . 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 3182 . . . . . . 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 3182 . . . . . 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 3216 . . . . . . 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 3208 . . 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 3279 . . 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 20477 . . . 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 20477 . . . 4 ((πœ‘ ∧ (𝑗 ∈ 𝐾 ∧ π‘˜ ∈ 𝐾)) β†’ (π‘˜ Β· π‘Œ) ∈ (π‘β€˜{π‘Œ}))
41 oveq1 7365 . . . . . 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 7366 . . . . . 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 3609 . . . 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 3208 . 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 3070  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  SubGrpcsubg 18927  LSSumclsm 19421  LModclmod 20336  LSpanclspn 20447
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-lsm 19423  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338  df-lss 20408  df-lsp 20448
This theorem is referenced by:  lsppr  20569  baerlem3lem2  40219  baerlem5alem2  40220  baerlem5blem2  40221
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