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Theorem lspprabs 20943
Description: Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)
Hypotheses
Ref Expression
lspprabs.v 𝑉 = (Baseβ€˜π‘Š)
lspprabs.p + = (+gβ€˜π‘Š)
lspprabs.n 𝑁 = (LSpanβ€˜π‘Š)
lspprabs.w (πœ‘ β†’ π‘Š ∈ LMod)
lspprabs.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
lspprabs.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
lspprabs (πœ‘ β†’ (π‘β€˜{𝑋, (𝑋 + π‘Œ)}) = (π‘β€˜{𝑋, π‘Œ}))

Proof of Theorem lspprabs
StepHypRef Expression
1 lspprabs.w . . . . . . 7 (πœ‘ β†’ π‘Š ∈ LMod)
2 eqid 2726 . . . . . . . 8 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
32lsssssubg 20805 . . . . . . 7 (π‘Š ∈ LMod β†’ (LSubSpβ€˜π‘Š) βŠ† (SubGrpβ€˜π‘Š))
41, 3syl 17 . . . . . 6 (πœ‘ β†’ (LSubSpβ€˜π‘Š) βŠ† (SubGrpβ€˜π‘Š))
5 lspprabs.x . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
6 lspprabs.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
7 lspprabs.n . . . . . . . 8 𝑁 = (LSpanβ€˜π‘Š)
86, 2, 7lspsncl 20824 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
91, 5, 8syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
104, 9sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
11 lspprabs.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
126, 2, 7lspsncl 20824 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
131, 11, 12syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
144, 13sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
15 eqid 2726 . . . . . 6 (LSSumβ€˜π‘Š) = (LSSumβ€˜π‘Š)
1615lsmub1 19577 . . . . 5 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š)) β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
1710, 14, 16syl2anc 583 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
182, 15lsmcl 20931 . . . . . 6 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š) ∧ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š)) β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (LSubSpβ€˜π‘Š))
191, 9, 13, 18syl3anc 1368 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (LSubSpβ€˜π‘Š))
206, 7lspsnid 20840 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
211, 5, 20syl2anc 583 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
226, 7lspsnid 20840 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ (π‘β€˜{π‘Œ}))
231, 11, 22syl2anc 583 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{π‘Œ}))
24 lspprabs.p . . . . . . 7 + = (+gβ€˜π‘Š)
2524, 15lsmelvali 19570 . . . . . 6 ((((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š)) ∧ (𝑋 ∈ (π‘β€˜{𝑋}) ∧ π‘Œ ∈ (π‘β€˜{π‘Œ}))) β†’ (𝑋 + π‘Œ) ∈ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
2610, 14, 21, 23, 25syl22anc 836 . . . . 5 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
272, 7, 1, 19, 26lspsnel5a 20843 . . . 4 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
286, 24lmodvacl 20721 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ 𝑉)
291, 5, 11, 28syl3anc 1368 . . . . . . 7 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝑉)
306, 2, 7lspsncl 20824 . . . . . . 7 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (LSubSpβ€˜π‘Š))
311, 29, 30syl2anc 583 . . . . . 6 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (LSubSpβ€˜π‘Š))
324, 31sseldd 3978 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (SubGrpβ€˜π‘Š))
334, 19sseldd 3978 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (SubGrpβ€˜π‘Š))
3415lsmlub 19584 . . . . 5 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (SubGrpβ€˜π‘Š) ∧ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (SubGrpβ€˜π‘Š)) β†’ (((π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∧ (π‘β€˜{(𝑋 + π‘Œ)}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ}))) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ}))))
3510, 32, 33, 34syl3anc 1368 . . . 4 (πœ‘ β†’ (((π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∧ (π‘β€˜{(𝑋 + π‘Œ)}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ}))) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ}))))
3617, 27, 35mpbi2and 709 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
3715lsmub1 19577 . . . . 5 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (SubGrpβ€˜π‘Š)) β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
3810, 32, 37syl2anc 583 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
392, 15lsmcl 20931 . . . . . 6 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š) ∧ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (LSubSpβ€˜π‘Š)) β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (LSubSpβ€˜π‘Š))
401, 9, 31, 39syl3anc 1368 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (LSubSpβ€˜π‘Š))
41 eqid 2726 . . . . . . 7 (-gβ€˜π‘Š) = (-gβ€˜π‘Š)
426, 7lspsnid 20840 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{(𝑋 + π‘Œ)}))
431, 29, 42syl2anc 583 . . . . . . 7 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{(𝑋 + π‘Œ)}))
4441, 15, 32, 10, 43, 21lsmelvalmi 19572 . . . . . 6 (πœ‘ β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) ∈ ((π‘β€˜{(𝑋 + π‘Œ)})(LSSumβ€˜π‘Š)(π‘β€˜{𝑋})))
45 lmodabl 20755 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Abel)
461, 45syl 17 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Abel)
476, 24, 41ablpncan2 19735 . . . . . . 7 ((π‘Š ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) = π‘Œ)
4846, 5, 11, 47syl3anc 1368 . . . . . 6 (πœ‘ β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) = π‘Œ)
4915lsmcom 19778 . . . . . . 7 ((π‘Š ∈ Abel ∧ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š)) β†’ ((π‘β€˜{(𝑋 + π‘Œ)})(LSSumβ€˜π‘Š)(π‘β€˜{𝑋})) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
5046, 32, 10, 49syl3anc 1368 . . . . . 6 (πœ‘ β†’ ((π‘β€˜{(𝑋 + π‘Œ)})(LSSumβ€˜π‘Š)(π‘β€˜{𝑋})) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
5144, 48, 503eltr3d 2841 . . . . 5 (πœ‘ β†’ π‘Œ ∈ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
522, 7, 1, 40, 51lspsnel5a 20843 . . . 4 (πœ‘ β†’ (π‘β€˜{π‘Œ}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
534, 40sseldd 3978 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (SubGrpβ€˜π‘Š))
5415lsmlub 19584 . . . . 5 (((π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š) ∧ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (SubGrpβ€˜π‘Š)) β†’ (((π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∧ (π‘β€˜{π‘Œ}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)}))) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)}))))
5510, 14, 53, 54syl3anc 1368 . . . 4 (πœ‘ β†’ (((π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∧ (π‘β€˜{π‘Œ}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)}))) ↔ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)}))))
5638, 52, 55mpbi2and 709 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
5736, 56eqssd 3994 . 2 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
586, 7, 15, 1, 5, 29lsmpr 20937 . 2 (πœ‘ β†’ (π‘β€˜{𝑋, (𝑋 + π‘Œ)}) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
596, 7, 15, 1, 5, 11lsmpr 20937 . 2 (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
6057, 58, 593eqtr4d 2776 1 (πœ‘ β†’ (π‘β€˜{𝑋, (𝑋 + π‘Œ)}) = (π‘β€˜{𝑋, π‘Œ}))
Colors of variables: wff setvar class
Syntax hints:   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  +gcplusg 17206  -gcsg 18865  SubGrpcsubg 19047  LSSumclsm 19554  Abelcabl 19701  LModclmod 20706  LSubSpclss 20778  LSpanclspn 20818
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
This theorem is referenced by:  lspabs2  20971  lspindp4  20988  mapdindp4  41107
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