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Theorem dvhopN 29995
 Description: Decompose a vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of and the other from the one-dimensional vector subspace . Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by , , . We swapped the order of vector sum (their juxtaposition i.e. composition) to show first. Note that and are the zero and one of the division ring , and is the zero of the translation group. is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhop.b
dvhop.h
dvhop.t
dvhop.e
dvhop.p
dvhop.a
dvhop.s
dvhop.o
Assertion
Ref Expression
dvhopN
Distinct variable groups:   ,   ,,,,,   ,   ,   ,,   ,,,,,,   ,,,
Allowed substitution hints:   (,,,,,)   (,,,,)   (,,,)   (,,,,,)   (,,,,,)   ()   (,,,,,)   (,,,,)   (,,,,)   (,,,,,)   (,,)

Proof of Theorem dvhopN
StepHypRef Expression
1 simprr 736 . . . . 5
2 dvhop.b . . . . . . 7
3 dvhop.h . . . . . . 7
4 dvhop.t . . . . . . 7
52, 3, 4idltrn 29028 . . . . . 6
65adantr 453 . . . . 5
7 dvhop.e . . . . . . 7
83, 4, 7tendoidcl 29647 . . . . . 6
98adantr 453 . . . . 5
10 dvhop.s . . . . . 6
1110dvhopspN 29994 . . . . 5
121, 6, 9, 11syl12anc 1185 . . . 4
132, 3, 7tendoid 29651 . . . . . 6
1413adantrl 699 . . . . 5
153, 4, 7tendo1mulr 29649 . . . . . 6
1615adantrl 699 . . . . 5
1714, 16opeq12d 3704 . . . 4
1812, 17eqtrd 2285 . . 3
1918oveq2d 5726 . 2
20 simprl 735 . . 3
21 dvhop.o . . . . 5
222, 3, 4, 7, 21tendo0cl 29668 . . . 4
2322adantr 453 . . 3
24 dvhop.a . . . 4
2524dvhopaddN 29993 . . 3
2620, 23, 6, 1, 25syl22anc 1188 . 2
272, 3, 4ltrn1o 29002 . . . . 5
2827adantrr 700 . . . 4
29 f1of 5329 . . . 4
30 fcoi1 5272 . . . 4
3128, 29, 303syl 20 . . 3
32 dvhop.p . . . . 5
332, 3, 4, 7, 21, 32tendo0pl 29669 . . . 4
3433adantrl 699 . . 3
3531, 34opeq12d 3704 . 2
3619, 26, 353eqtrrd 2290 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1619   wcel 1621  cop 3547   cmpt 3974   cid 4197   cxp 4578   cres 4582   ccom 4584  wf 4588  wf1o 4591  cfv 4592  (class class class)co 5710   cmpt2 5712  c1st 5972  c2nd 5973  cbs 13022  chlt 28229  clh 28862  cltrn 28979  ctendo 29630 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-llines 28376  df-lplanes 28377  df-lvols 28378  df-lines 28379  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037  df-tendo 29633
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