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Theorem lspindp5 41154
Description: Obtain an independent vector set π‘ˆ, 𝑋, π‘Œ from a vector π‘ˆ dependent on 𝑋 and 𝑍 and another independent set 𝑍, 𝑋, π‘Œ. (Here we don't show the (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) part of the independence, which passes straight through. We also don't show nonzero vector requirements that are redundant for this theorem. Different orderings can be obtained using lspexch 20980 and prcom 4731.) (Contributed by NM, 4-May-2015.)
Hypotheses
Ref Expression
lspindp5.v 𝑉 = (Baseβ€˜π‘Š)
lspindp5.n 𝑁 = (LSpanβ€˜π‘Š)
lspindp5.w (πœ‘ β†’ π‘Š ∈ LVec)
lspindp5.y (πœ‘ β†’ 𝑋 ∈ 𝑉)
lspindp5.x (πœ‘ β†’ π‘Œ ∈ 𝑉)
lspindp5.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
lspindp5.e (πœ‘ β†’ 𝑍 ∈ (π‘β€˜{𝑋, π‘ˆ}))
lspindp5.m (πœ‘ β†’ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ}))
Assertion
Ref Expression
lspindp5 (πœ‘ β†’ Β¬ π‘ˆ ∈ (π‘β€˜{𝑋, π‘Œ}))
Proof of Theorem lspindp5
StepHypRef Expression
1 lspindp5.m . . 3 (πœ‘ β†’ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ}))
2 lspindp5.e . . . 4 (πœ‘ β†’ 𝑍 ∈ (π‘β€˜{𝑋, π‘ˆ}))
3 ssel 3970 . . . 4 ((π‘β€˜{𝑋, π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}) β†’ (𝑍 ∈ (π‘β€˜{𝑋, π‘ˆ}) β†’ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
42, 3syl5com 31 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋, π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}) β†’ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
51, 4mtod 197 . 2 (πœ‘ β†’ Β¬ (π‘β€˜{𝑋, π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}))
6 lspindp5.w . . . . . . 7 (πœ‘ β†’ π‘Š ∈ LVec)
7 lveclmod 20954 . . . . . . 7 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
86, 7syl 17 . . . . . 6 (πœ‘ β†’ π‘Š ∈ LMod)
9 lspindp5.y . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
10 lspindp5.x . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
11 prssi 4819 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ {𝑋, π‘Œ} βŠ† 𝑉)
129, 10, 11syl2anc 583 . . . . . 6 (πœ‘ β†’ {𝑋, π‘Œ} βŠ† 𝑉)
13 snsspr1 4812 . . . . . . 7 {𝑋} βŠ† {𝑋, π‘Œ}
1413a1i 11 . . . . . 6 (πœ‘ β†’ {𝑋} βŠ† {𝑋, π‘Œ})
15 lspindp5.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
16 lspindp5.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘Š)
1715, 16lspss 20831 . . . . . 6 ((π‘Š ∈ LMod ∧ {𝑋, π‘Œ} βŠ† 𝑉 ∧ {𝑋} βŠ† {𝑋, π‘Œ}) β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{𝑋, π‘Œ}))
188, 12, 14, 17syl3anc 1368 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{𝑋, π‘Œ}))
1918biantrurd 532 . . . 4 (πœ‘ β†’ ((π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ↔ ((π‘β€˜{𝑋}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ∧ (π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}))))
20 eqid 2726 . . . . . . . 8 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
2120lsssssubg 20805 . . . . . . 7 (π‘Š ∈ LMod β†’ (LSubSpβ€˜π‘Š) βŠ† (SubGrpβ€˜π‘Š))
228, 21syl 17 . . . . . 6 (πœ‘ β†’ (LSubSpβ€˜π‘Š) βŠ† (SubGrpβ€˜π‘Š))
2315, 20, 16lspsncl 20824 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
248, 9, 23syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
2522, 24sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
26 lspindp5.u . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
2715, 20, 16lspsncl 20824 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑉) β†’ (π‘β€˜{π‘ˆ}) ∈ (LSubSpβ€˜π‘Š))
288, 26, 27syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘β€˜{π‘ˆ}) ∈ (LSubSpβ€˜π‘Š))
2922, 28sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘ˆ}) ∈ (SubGrpβ€˜π‘Š))
3015, 20, 16, 8, 9, 10lspprcl 20825 . . . . . 6 (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
3122, 30sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
32 eqid 2726 . . . . . 6 (LSSumβ€˜π‘Š) = (LSSumβ€˜π‘Š)
3332lsmlub 19584 . . . . 5 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{π‘ˆ}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑋, π‘Œ}) ∈ (SubGrpβ€˜π‘Š)) β†’ (((π‘β€˜{𝑋}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ∧ (π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ})) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘ˆ})) βŠ† (π‘β€˜{𝑋, π‘Œ})))
3425, 29, 31, 33syl3anc 1368 . . . 4 (πœ‘ β†’ (((π‘β€˜{𝑋}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ∧ (π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ})) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘ˆ})) βŠ† (π‘β€˜{𝑋, π‘Œ})))
3519, 34bitrd 279 . . 3 (πœ‘ β†’ ((π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘ˆ})) βŠ† (π‘β€˜{𝑋, π‘Œ})))
3615, 20, 16, 8, 30, 26lspsnel5 20842 . . 3 (πœ‘ β†’ (π‘ˆ ∈ (π‘β€˜{𝑋, π‘Œ}) ↔ (π‘β€˜{π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ})))
3715, 16, 32, 8, 9, 26lsmpr 20937 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋, π‘ˆ}) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘ˆ})))
3837sseq1d 4008 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋, π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ}) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘ˆ})) βŠ† (π‘β€˜{𝑋, π‘Œ})))
3935, 36, 383bitr4d 311 . 2 (πœ‘ β†’ (π‘ˆ ∈ (π‘β€˜{𝑋, π‘Œ}) ↔ (π‘β€˜{𝑋, π‘ˆ}) βŠ† (π‘β€˜{𝑋, π‘Œ})))
405, 39mtbird 325 1 (πœ‘ β†’ Β¬ π‘ˆ ∈ (π‘β€˜{𝑋, π‘Œ}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  {csn 4623  {cpr 4625  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  SubGrpcsubg 19047  LSSumclsm 19554  LModclmod 20706  LSubSpclss 20778  LSpanclspn 20818  LVecclvec 20950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  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 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-cntz 19233  df-lsm 19556  df-cmn 19702  df-abl 19703  df-mgp 20040  df-ur 20087  df-ring 20140  df-lmod 20708  df-lss 20779  df-lsp 20819  df-lvec 20951
This theorem is referenced by:  mapdh8b  41164
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