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Theorem mapdh6lem1N 41238
Description: Lemma for mapdh6N 41252. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
mapdh.q 𝑄 = (0gβ€˜πΆ)
mapdh.i 𝐼 = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))
mapdh.h 𝐻 = (LHypβ€˜πΎ)
mapdh.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
mapdh.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
mapdh.v 𝑉 = (Baseβ€˜π‘ˆ)
mapdh.s βˆ’ = (-gβ€˜π‘ˆ)
mapdhc.o 0 = (0gβ€˜π‘ˆ)
mapdh.n 𝑁 = (LSpanβ€˜π‘ˆ)
mapdh.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
mapdh.d 𝐷 = (Baseβ€˜πΆ)
mapdh.r 𝑅 = (-gβ€˜πΆ)
mapdh.j 𝐽 = (LSpanβ€˜πΆ)
mapdh.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
mapdhc.f (πœ‘ β†’ 𝐹 ∈ 𝐷)
mapdh.mn (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))
mapdhcl.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
mapdh.p + = (+gβ€˜π‘ˆ)
mapdh.a ✚ = (+gβ€˜πΆ)
mapdhe6.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
mapdhe6.z (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
mapdhe6.xn (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
mapdh6.yz (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
mapdh6.fg (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
mapdh6.fe (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
Assertion
Ref Expression
mapdh6lem1N (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
Distinct variable groups:   π‘₯,𝐷,β„Ž   β„Ž,𝐹,π‘₯   π‘₯,𝐽   π‘₯,𝑀   π‘₯,𝑁   π‘₯, 0   π‘₯,𝑄   π‘₯,𝑅   π‘₯, βˆ’   β„Ž,𝑋,π‘₯   β„Ž,π‘Œ,π‘₯   πœ‘,β„Ž   0 ,β„Ž   𝐢,β„Ž   𝐷,β„Ž   β„Ž,𝐽   β„Ž,𝑀   β„Ž,𝑁   𝑅,β„Ž   π‘ˆ,β„Ž   βˆ’ ,β„Ž   β„Ž,𝐺,π‘₯   β„Ž,𝐸   β„Ž,𝑍,π‘₯   ✚ ,β„Ž   β„Ž,𝐼   + ,β„Ž,π‘₯
Allowed substitution hints:   πœ‘(π‘₯)   𝐢(π‘₯)   ✚ (π‘₯)   𝑄(β„Ž)   π‘ˆ(π‘₯)   𝐸(π‘₯)   𝐻(π‘₯,β„Ž)   𝐼(π‘₯)   𝐾(π‘₯,β„Ž)   𝑉(π‘₯,β„Ž)   π‘Š(π‘₯,β„Ž)

Proof of Theorem mapdh6lem1N
StepHypRef Expression
1 mapdh.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 mapdh.m . . . 4 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
3 mapdh.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
4 eqid 2728 . . . 4 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
5 mapdh.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
61, 3, 5dvhlmod 40615 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 mapdhcl.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
87eldifad 3961 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
9 mapdhe6.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
109eldifad 3961 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
11 mapdh.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘ˆ)
12 mapdh.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
1311, 12lmodvsubcl 20797 . . . . . . 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 mapdh.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘ˆ)
1611, 4, 15lspsncl 20868 . . . . . 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 mapdhe6.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1918eldifad 3961 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
2011, 4, 15lspsncl 20868 . . . . . 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 2728 . . . . . 6 (LSSumβ€˜π‘ˆ) = (LSSumβ€˜π‘ˆ)
234, 22lsmcl 20975 . . . . 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 20797 . . . . . . 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 20868 . . . . . 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 20868 . . . . . 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 20975 . . . . 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 41167 . . 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 mapdh.c . . . . . 6 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
35 eqid 2728 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
361, 2, 3, 4, 22, 34, 35, 5, 17, 21mapdlsm 41169 . . . . 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 41169 . . . . 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 4215 . . . 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 mapdh6.fg . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
40 mapdh.q . . . . . . . . 9 𝑄 = (0gβ€˜πΆ)
41 mapdh.i . . . . . . . . 9 𝐼 = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))
42 mapdhc.o . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
43 mapdh.d . . . . . . . . 9 𝐷 = (Baseβ€˜πΆ)
44 mapdh.r . . . . . . . . 9 𝑅 = (-gβ€˜πΆ)
45 mapdh.j . . . . . . . . 9 𝐽 = (LSpanβ€˜πΆ)
46 mapdhc.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ 𝐷)
47 mapdh.mn . . . . . . . . 9 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))
481, 3, 5dvhlvec 40614 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LVec)
49 mapdh6.yz . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
50 mapdhe6.xn . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
5111, 42, 15, 48, 10, 18, 8, 49, 50lspindp2 21030 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
5251simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
5340, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 10, 52mapdhcl 41232 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5439, 53eqeltrrd 2830 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ 𝐷)
5540, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 9, 54, 52mapdheq 41233 . . . . . . . 8 (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺 ↔ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐹𝑅𝐺)}))))
5639, 55mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐹𝑅𝐺)})))
5756simprd 494 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐹𝑅𝐺)}))
58 mapdh6.fe . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
5911, 42, 15, 48, 9, 19, 8, 49, 50lspindp1 21028 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) ∧ Β¬ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
6059simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
6140, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 19, 60mapdhcl 41232 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6258, 61eqeltrrd 2830 . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ 𝐷)
6340, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 18, 62, 60mapdheq 41233 . . . . . . . 8 (πœ‘ β†’ ((πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸 ↔ ((π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)}))))
6458, 63mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)})))
6564simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}))
6657, 65oveq12d 7444 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) = ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{𝐸})))
6764simprd 494 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)}))
6856simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))
6967, 68oveq12d 7444 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{π‘Œ}))) = ((π½β€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(π½β€˜{𝐺})))
7066, 69ineq12d 4215 . . . 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½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(π½β€˜{𝐺}))))
7138, 70eqtrd 2768 . . 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½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(π½β€˜{𝐺}))))
7233, 71eqtrd 2768 . 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½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(π½β€˜{𝐺}))))
73 mapdh.p . . . 4 + = (+gβ€˜π‘ˆ)
7411, 12, 42, 22, 15, 48, 8, 50, 49, 9, 18, 73baerlem5a 41219 . . 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}))))
7574fveq2d 6906 . 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})))))
761, 34, 5lcdlvec 41096 . . 3 (πœ‘ β†’ 𝐢 ∈ LVec)
771, 2, 3, 11, 15, 34, 43, 45, 5, 46, 47, 8, 10, 54, 68, 19, 62, 65, 50mapdindp 41176 . . 3 (πœ‘ β†’ Β¬ 𝐹 ∈ (π½β€˜{𝐺, 𝐸}))
781, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 10, 19, 62, 65, 49mapdncol 41175 . . 3 (πœ‘ β†’ (π½β€˜{𝐺}) β‰  (π½β€˜{𝐸}))
791, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 42, 40, 9mapdn0 41174 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {𝑄}))
801, 2, 3, 11, 15, 34, 43, 45, 5, 62, 65, 42, 40, 18mapdn0 41174 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {𝑄}))
81 mapdh.a . . 3 ✚ = (+gβ€˜πΆ)
8243, 44, 40, 35, 45, 76, 46, 77, 78, 79, 80, 81baerlem5a 41219 . 2 (πœ‘ β†’ (π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐸)})(LSSumβ€˜πΆ)(π½β€˜{𝐺}))))
8372, 75, 823eqtr4d 2778 1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (π½β€˜{(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 2937  Vcvv 3473   βˆ– cdif 3946   ∩ cin 3948  ifcif 4532  {csn 4632  {cpr 4634  βŸ¨cotp 4640   ↦ cmpt 5235  β€˜cfv 6553  β„©crio 7381  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  Basecbs 17187  +gcplusg 17240  0gc0g 17428  -gcsg 18899  LSSumclsm 19596  LModclmod 20750  LSubSpclss 20822  LSpanclspn 20862  HLchlt 38854  LHypclh 39489  DVecHcdvh 40583  LCDualclcd 41091  mapdcmpd 41129
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-riotaBAD 38457
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-ot 4641  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-undef 8285  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-0g 17430  df-mre 17573  df-mrc 17574  df-acs 17576  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-cntz 19275  df-oppg 19304  df-lsm 19598  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-dvr 20347  df-drng 20633  df-lmod 20752  df-lss 20823  df-lsp 20863  df-lvec 20995  df-lsatoms 38480  df-lshyp 38481  df-lcv 38523  df-lfl 38562  df-lkr 38590  df-ldual 38628  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-tgrp 40248  df-tendo 40260  df-edring 40262  df-dveca 40508  df-disoa 40534  df-dvech 40584  df-dib 40644  df-dic 40678  df-dih 40734  df-doch 40853  df-djh 40900  df-lcdual 41092  df-mapd 41130
This theorem is referenced by:  mapdh6lem2N  41239  mapdh6aN  41240
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