Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1l6lem1 Structured version   Visualization version   GIF version

Theorem hdmap1l6lem1 41308
Description: Lemma for hdmap1l6 41322. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.)
Hypotheses
Ref Expression
hdmap1l6.h 𝐻 = (LHypβ€˜πΎ)
hdmap1l6.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1l6.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1l6.p + = (+gβ€˜π‘ˆ)
hdmap1l6.s βˆ’ = (-gβ€˜π‘ˆ)
hdmap1l6c.o 0 = (0gβ€˜π‘ˆ)
hdmap1l6.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1l6.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1l6.d 𝐷 = (Baseβ€˜πΆ)
hdmap1l6.a ✚ = (+gβ€˜πΆ)
hdmap1l6.r 𝑅 = (-gβ€˜πΆ)
hdmap1l6.q 𝑄 = (0gβ€˜πΆ)
hdmap1l6.l 𝐿 = (LSpanβ€˜πΆ)
hdmap1l6.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1l6.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1l6.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
hdmap1l6.f (πœ‘ β†’ 𝐹 ∈ 𝐷)
hdmap1l6cl.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
hdmap1l6.mn (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (πΏβ€˜{𝐹}))
hdmap1l6e.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
hdmap1l6e.z (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
hdmap1l6e.xn (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
hdmap1l6.yz (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
hdmap1l6.fg (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
hdmap1l6.fe (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
Assertion
Ref Expression
hdmap1l6lem1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Proof of Theorem hdmap1l6lem1
StepHypRef Expression
1 hdmap1l6.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 hdmap1l6.m . . . 4 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
3 hdmap1l6.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
4 eqid 2725 . . . 4 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
5 hdmap1l6.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
61, 3, 5dvhlmod 40611 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 hdmap1l6cl.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
87eldifad 3951 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
9 hdmap1l6e.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
109eldifad 3951 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
11 hdmap1l6.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘ˆ)
12 hdmap1l6.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
1311, 12lmodvsubcl 20792 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 βˆ’ π‘Œ) ∈ 𝑉)
146, 8, 10, 13syl3anc 1368 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) ∈ 𝑉)
15 hdmap1l6.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘ˆ)
1611, 4, 15lspsncl 20863 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ π‘Œ) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ))
176, 14, 16syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ))
18 hdmap1l6e.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1918eldifad 3951 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
2011, 4, 15lspsncl 20863 . . . . . 6 ((π‘ˆ ∈ LMod ∧ 𝑍 ∈ 𝑉) β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
216, 19, 20syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
22 eqid 2725 . . . . . 6 (LSSumβ€˜π‘ˆ) = (LSSumβ€˜π‘ˆ)
234, 22lsmcl 20970 . . . . 5 ((π‘ˆ ∈ LMod ∧ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ) ∧ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ)) β†’ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∈ (LSubSpβ€˜π‘ˆ))
246, 17, 21, 23syl3anc 1368 . . . 4 (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∈ (LSubSpβ€˜π‘ˆ))
2511, 12lmodvsubcl 20792 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) β†’ (𝑋 βˆ’ 𝑍) ∈ 𝑉)
266, 8, 19, 25syl3anc 1368 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ 𝑍) ∈ 𝑉)
2711, 4, 15lspsncl 20863 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ 𝑍) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ))
286, 26, 27syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ))
2911, 4, 15lspsncl 20863 . . . . . 6 ((π‘ˆ ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
306, 10, 29syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
314, 22lsmcl 20970 . . . . 5 ((π‘ˆ ∈ LMod ∧ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ) ∧ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ)) β†’ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})) ∈ (LSubSpβ€˜π‘ˆ))
326, 28, 30, 31syl3anc 1368 . . . 4 (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})) ∈ (LSubSpβ€˜π‘ˆ))
331, 2, 3, 4, 5, 24, 32mapdin 41163 . . 3 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))) = ((π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))))
34 hdmap1l6.c . . . . . 6 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
35 eqid 2725 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
361, 2, 3, 4, 22, 34, 35, 5, 17, 21mapdlsm 41165 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) = ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))))
371, 2, 3, 4, 22, 34, 35, 5, 28, 30mapdlsm 41165 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ}))) = ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{π‘Œ}))))
3836, 37ineq12d 4205 . . . 4 (πœ‘ β†’ ((π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))) = (((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) ∩ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{π‘Œ})))))
39 hdmap1l6.fg . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
40 hdmap1l6c.o . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
41 hdmap1l6.d . . . . . . . . 9 𝐷 = (Baseβ€˜πΆ)
42 hdmap1l6.r . . . . . . . . 9 𝑅 = (-gβ€˜πΆ)
43 hdmap1l6.l . . . . . . . . 9 𝐿 = (LSpanβ€˜πΆ)
44 hdmap1l6.i . . . . . . . . 9 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
45 hdmap1l6.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ 𝐷)
46 hdmap1l6.mn . . . . . . . . . . 11 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (πΏβ€˜{𝐹}))
471, 3, 5dvhlvec 40610 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LVec)
48 hdmap1l6.yz . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
49 hdmap1l6e.xn . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
5011, 40, 15, 47, 10, 18, 8, 48, 49lspindp2 21025 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
5150simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
521, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 51, 7, 10hdmap1cl 41305 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5339, 52eqeltrrd 2826 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ 𝐷)
541, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 9, 53, 51, 46hdmap1eq 41302 . . . . . . . 8 (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺 ↔ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (πΏβ€˜{𝐺}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (πΏβ€˜{(𝐹𝑅𝐺)}))))
5539, 54mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (πΏβ€˜{𝐺}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (πΏβ€˜{(𝐹𝑅𝐺)})))
5655simprd 494 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (πΏβ€˜{(𝐹𝑅𝐺)}))
57 hdmap1l6.fe . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
5811, 40, 15, 47, 9, 19, 8, 48, 49lspindp1 21023 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) ∧ Β¬ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
5958simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
601, 3, 11, 40, 15, 34, 41, 43, 2, 44, 5, 45, 46, 59, 7, 19hdmap1cl 41305 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6157, 60eqeltrrd 2826 . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ 𝐷)
621, 3, 11, 12, 40, 15, 34, 41, 42, 43, 2, 44, 5, 7, 45, 18, 61, 59, 46hdmap1eq 41302 . . . . . . . 8 (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸 ↔ ((π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (πΏβ€˜{(𝐹𝑅𝐸)}))))
6357, 62mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (πΏβ€˜{(𝐹𝑅𝐸)})))
6463simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}))
6556, 64oveq12d 7432 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) = ((πΏβ€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})))
6663simprd 494 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (πΏβ€˜{(𝐹𝑅𝐸)}))
6755simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (πΏβ€˜{𝐺}))
6866, 67oveq12d 7432 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{π‘Œ}))) = ((πΏβ€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐺})))
6965, 68ineq12d 4205 . . . 4 (πœ‘ β†’ (((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) ∩ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{π‘Œ})))) = (((πΏβ€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐺}))))
7038, 69eqtrd 2765 . . 3 (πœ‘ β†’ ((π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))) = (((πΏβ€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐺}))))
7133, 70eqtrd 2765 . 2 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))) = (((πΏβ€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐺}))))
72 hdmap1l6.p . . . 4 + = (+gβ€˜π‘ˆ)
7311, 12, 40, 22, 15, 47, 8, 49, 48, 9, 18, 72baerlem5a 41215 . . 3 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) = (((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ}))))
7473fveq2d 6894 . 2 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (π‘€β€˜(((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ 𝑍)})(LSSumβ€˜π‘ˆ)(π‘β€˜{π‘Œ})))))
75 hdmap1l6.q . . 3 𝑄 = (0gβ€˜πΆ)
761, 34, 5lcdlvec 41092 . . 3 (πœ‘ β†’ 𝐢 ∈ LVec)
771, 2, 3, 11, 15, 34, 41, 43, 5, 45, 46, 8, 10, 53, 67, 19, 61, 64, 49mapdindp 41172 . . 3 (πœ‘ β†’ Β¬ 𝐹 ∈ (πΏβ€˜{𝐺, 𝐸}))
781, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 10, 19, 61, 64, 48mapdncol 41171 . . 3 (πœ‘ β†’ (πΏβ€˜{𝐺}) β‰  (πΏβ€˜{𝐸}))
791, 2, 3, 11, 15, 34, 41, 43, 5, 53, 67, 40, 75, 9mapdn0 41170 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {𝑄}))
801, 2, 3, 11, 15, 34, 41, 43, 5, 61, 64, 40, 75, 18mapdn0 41170 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {𝑄}))
81 hdmap1l6.a . . 3 ✚ = (+gβ€˜πΆ)
8241, 42, 75, 35, 43, 76, 45, 77, 78, 79, 80, 81baerlem5a 41215 . 2 (πœ‘ β†’ (πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((πΏβ€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(πΏβ€˜{𝐺}))))
8371, 74, 823eqtr4d 2775 1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   βˆ– cdif 3936   ∩ cin 3938  {csn 4622  {cpr 4624  βŸ¨cotp 4630  β€˜cfv 6541  (class class class)co 7414  Basecbs 17177  +gcplusg 17230  0gc0g 17418  -gcsg 18894  LSSumclsm 19591  LModclmod 20745  LSubSpclss 20817  LSpanclspn 20857  HLchlt 38850  LHypclh 39485  DVecHcdvh 40579  LCDualclcd 41087  mapdcmpd 41125  HDMap1chdma1 41292
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 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-riotaBAD 38453
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4943  df-iun 4991  df-iin 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-om 7867  df-1st 7989  df-2nd 7990  df-tpos 8228  df-undef 8275  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-sca 17246  df-vsca 17247  df-0g 17420  df-mre 17563  df-mrc 17564  df-acs 17566  df-proset 18284  df-poset 18302  df-plt 18319  df-lub 18335  df-glb 18336  df-join 18337  df-meet 18338  df-p0 18414  df-p1 18415  df-lat 18421  df-clat 18488  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-submnd 18738  df-grp 18895  df-minusg 18896  df-sbg 18897  df-subg 19080  df-cntz 19270  df-oppg 19299  df-lsm 19593  df-cmn 19739  df-abl 19740  df-mgp 20077  df-rng 20095  df-ur 20124  df-ring 20177  df-oppr 20275  df-dvdsr 20298  df-unit 20299  df-invr 20329  df-dvr 20342  df-drng 20628  df-lmod 20747  df-lss 20818  df-lsp 20858  df-lvec 20990  df-lsatoms 38476  df-lshyp 38477  df-lcv 38519  df-lfl 38558  df-lkr 38586  df-ldual 38624  df-oposet 38676  df-ol 38678  df-oml 38679  df-covers 38766  df-ats 38767  df-atl 38798  df-cvlat 38822  df-hlat 38851  df-llines 38999  df-lplanes 39000  df-lvols 39001  df-lines 39002  df-psubsp 39004  df-pmap 39005  df-padd 39297  df-lhyp 39489  df-laut 39490  df-ldil 39605  df-ltrn 39606  df-trl 39660  df-tgrp 40244  df-tendo 40256  df-edring 40258  df-dveca 40504  df-disoa 40530  df-dvech 40580  df-dib 40640  df-dic 40674  df-dih 40730  df-doch 40849  df-djh 40896  df-lcdual 41088  df-mapd 41126  df-hdmap1 41294
This theorem is referenced by:  hdmap1l6lem2  41309  hdmap1l6a  41310
  Copyright terms: Public domain W3C validator