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Theorem lspprabs 20572
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 2737 . . . . . . . 8 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
32lsssssubg 20435 . . . . . . 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 20454 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
91, 5, 8syl2anc 585 . . . . . 6 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘Š))
104, 9sseldd 3950 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
11 lspprabs.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
126, 2, 7lspsncl 20454 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
131, 11, 12syl2anc 585 . . . . . 6 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
144, 13sseldd 3950 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (SubGrpβ€˜π‘Š))
15 eqid 2737 . . . . . 6 (LSSumβ€˜π‘Š) = (LSSumβ€˜π‘Š)
1615lsmub1 19446 . . . . 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 585 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
182, 15lsmcl 20560 . . . . . 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 1372 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (LSubSpβ€˜π‘Š))
206, 7lspsnid 20470 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
211, 5, 20syl2anc 585 . . . . . 6 (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
226, 7lspsnid 20470 . . . . . . 7 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ (π‘β€˜{π‘Œ}))
231, 11, 22syl2anc 585 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{π‘Œ}))
24 lspprabs.p . . . . . . 7 + = (+gβ€˜π‘Š)
2524, 15lsmelvali 19439 . . . . . 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 838 . . . . 5 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
272, 7, 1, 19, 26lspsnel5a 20473 . . . 4 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
286, 24lmodvacl 20352 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ 𝑉)
291, 5, 11, 28syl3anc 1372 . . . . . . 7 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ 𝑉)
306, 2, 7lspsncl 20454 . . . . . . 7 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (LSubSpβ€˜π‘Š))
311, 29, 30syl2anc 585 . . . . . 6 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (LSubSpβ€˜π‘Š))
324, 31sseldd 3950 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 + π‘Œ)}) ∈ (SubGrpβ€˜π‘Š))
334, 19sseldd 3950 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})) ∈ (SubGrpβ€˜π‘Š))
3415lsmlub 19453 . . . . 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 1372 . . . 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 711 . . 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 19446 . . . . 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 585 . . . 4 (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
392, 15lsmcl 20560 . . . . . 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 1372 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (LSubSpβ€˜π‘Š))
41 eqid 2737 . . . . . . 7 (-gβ€˜π‘Š) = (-gβ€˜π‘Š)
426, 7lspsnid 20470 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{(𝑋 + π‘Œ)}))
431, 29, 42syl2anc 585 . . . . . . 7 (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{(𝑋 + π‘Œ)}))
4441, 15, 32, 10, 43, 21lsmelvalmi 19441 . . . . . 6 (πœ‘ β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) ∈ ((π‘β€˜{(𝑋 + π‘Œ)})(LSSumβ€˜π‘Š)(π‘β€˜{𝑋})))
45 lmodabl 20385 . . . . . . . 8 (π‘Š ∈ LMod β†’ π‘Š ∈ Abel)
461, 45syl 17 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ Abel)
476, 24, 41ablpncan2 19601 . . . . . . 7 ((π‘Š ∈ Abel ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) = π‘Œ)
4846, 5, 11, 47syl3anc 1372 . . . . . 6 (πœ‘ β†’ ((𝑋 + π‘Œ)(-gβ€˜π‘Š)𝑋) = π‘Œ)
4915lsmcom 19643 . . . . . . 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 1372 . . . . . 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 2852 . . . . 5 (πœ‘ β†’ π‘Œ ∈ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
522, 7, 1, 40, 51lspsnel5a 20473 . . . 4 (πœ‘ β†’ (π‘β€˜{π‘Œ}) βŠ† ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})))
534, 40sseldd 3950 . . . . 5 (πœ‘ β†’ ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{(𝑋 + π‘Œ)})) ∈ (SubGrpβ€˜π‘Š))
5415lsmlub 19453 . . . . 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 1372 . . . 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 711 . . 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 3966 . 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 20566 . 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 20566 . 2 (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) = ((π‘β€˜{𝑋})(LSSumβ€˜π‘Š)(π‘β€˜{π‘Œ})))
6057, 58, 593eqtr4d 2787 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3915  {csn 4591  {cpr 4593  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  -gcsg 18757  SubGrpcsubg 18929  LSSumclsm 19423  Abelcabl 19570  LModclmod 20338  LSubSpclss 20408  LSpanclspn 20448
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-lmod 20340  df-lss 20409  df-lsp 20449
This theorem is referenced by:  lspabs2  20597  lspindp4  20614  mapdindp4  40215
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