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Theorem mapdheq4lem 40244
Description: Lemma for mapdheq4 40245. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.)
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 }))
mapdhe4.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
mapdhe.z (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
mapdh.xn (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
mapdh.yz (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
mapdh.eg (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
mapdh.ee (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
Assertion
Ref Expression
mapdheq4lem (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ βˆ’ 𝑍)})) = (π½β€˜{(𝐺𝑅𝐸)}))
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‘₯
Allowed substitution hints:   πœ‘(π‘₯)   𝐢(π‘₯)   𝑄(β„Ž)   π‘ˆ(π‘₯)   𝐸(π‘₯)   𝐻(π‘₯,β„Ž)   𝐼(π‘₯,β„Ž)   𝐾(π‘₯,β„Ž)   𝑉(π‘₯,β„Ž)   π‘Š(π‘₯,β„Ž)

Proof of Theorem mapdheq4lem
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 2733 . . . 4 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
5 mapdh.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
61, 3, 5dvhlmod 39623 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 mapdhe4.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
87eldifad 3926 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑉)
9 mapdh.v . . . . . . 7 𝑉 = (Baseβ€˜π‘ˆ)
10 mapdh.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘ˆ)
119, 4, 10lspsncl 20482 . . . . . 6 ((π‘ˆ ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
126, 8, 11syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
13 mapdhe.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1413eldifad 3926 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
159, 4, 10lspsncl 20482 . . . . . 6 ((π‘ˆ ∈ LMod ∧ 𝑍 ∈ 𝑉) β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
166, 14, 15syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
17 eqid 2733 . . . . . 6 (LSSumβ€˜π‘ˆ) = (LSSumβ€˜π‘ˆ)
184, 17lsmcl 20588 . . . . 5 ((π‘ˆ ∈ LMod ∧ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ) ∧ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ)) β†’ ((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∈ (LSubSpβ€˜π‘ˆ))
196, 12, 16, 18syl3anc 1372 . . . 4 (πœ‘ β†’ ((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∈ (LSubSpβ€˜π‘ˆ))
20 mapdhcl.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2120eldifad 3926 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
22 mapdh.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
239, 22lmodvsubcl 20411 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 βˆ’ π‘Œ) ∈ 𝑉)
246, 21, 8, 23syl3anc 1372 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) ∈ 𝑉)
259, 4, 10lspsncl 20482 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ π‘Œ) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ))
266, 24, 25syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ))
279, 22lmodvsubcl 20411 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) β†’ (𝑋 βˆ’ 𝑍) ∈ 𝑉)
286, 21, 14, 27syl3anc 1372 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ 𝑍) ∈ 𝑉)
299, 4, 10lspsncl 20482 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ 𝑍) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ))
306, 28, 29syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ))
314, 17lsmcl 20588 . . . . 5 ((π‘ˆ ∈ LMod ∧ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) ∈ (LSubSpβ€˜π‘ˆ) ∧ (π‘β€˜{(𝑋 βˆ’ 𝑍)}) ∈ (LSubSpβ€˜π‘ˆ)) β†’ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})) ∈ (LSubSpβ€˜π‘ˆ))
326, 26, 30, 31syl3anc 1372 . . . 4 (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})) ∈ (LSubSpβ€˜π‘ˆ))
331, 2, 3, 4, 5, 19, 32mapdin 40175 . . 3 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})))) = ((π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})))))
34 mapdh.c . . . . . 6 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
35 eqid 2733 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
361, 2, 3, 4, 17, 34, 35, 5, 12, 16mapdlsm 40177 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) = ((π‘€β€˜(π‘β€˜{π‘Œ}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))))
37 mapdh.eg . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
38 mapdh.q . . . . . . . . 9 𝑄 = (0gβ€˜πΆ)
39 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)})))))
40 mapdhc.o . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
41 mapdh.d . . . . . . . . 9 𝐷 = (Baseβ€˜πΆ)
42 mapdh.r . . . . . . . . 9 𝑅 = (-gβ€˜πΆ)
43 mapdh.j . . . . . . . . 9 𝐽 = (LSpanβ€˜πΆ)
44 mapdhc.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ 𝐷)
45 mapdh.mn . . . . . . . . 9 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))
461, 3, 5dvhlvec 39622 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LVec)
47 mapdh.yz . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
48 mapdh.xn . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
499, 40, 10, 46, 8, 13, 21, 47, 48lspindp2 20641 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
5049simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
5138, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 8, 50mapdhcl 40240 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5237, 51eqeltrrd 2835 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ 𝐷)
5338, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 7, 52, 50mapdheq 40241 . . . . . . . 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)}))))
5437, 53mpbid 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)})))
5554simpld 496 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))
56 mapdh.ee . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
579, 40, 10, 46, 7, 14, 21, 47, 48lspindp1 20639 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) ∧ Β¬ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
5857simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
5938, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 14, 58mapdhcl 40240 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6056, 59eqeltrrd 2835 . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ 𝐷)
6138, 39, 1, 2, 3, 9, 22, 40, 10, 34, 41, 42, 43, 5, 44, 45, 20, 13, 60, 58mapdheq 40241 . . . . . . . 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)}))))
6256, 61mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)})))
6362simpld 496 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}))
6455, 63oveq12d 7379 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{π‘Œ}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) = ((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})))
6536, 64eqtrd 2773 . . . 4 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) = ((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})))
661, 2, 3, 4, 17, 34, 35, 5, 26, 30mapdlsm 40177 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)}))) = ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))))
6754simprd 497 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐹𝑅𝐺)}))
6862simprd 497 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)}))
6967, 68oveq12d 7379 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)}))) = ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{(𝐹𝑅𝐸)})))
7066, 69eqtrd 2773 . . . 4 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)}))) = ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{(𝐹𝑅𝐸)})))
7165, 70ineq12d 4177 . . 3 (πœ‘ β†’ ((π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})))) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{(𝐹𝑅𝐸)}))))
7233, 71eqtrd 2773 . 2 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})))) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{(𝐹𝑅𝐸)}))))
739, 22, 40, 17, 10, 46, 21, 48, 47, 7, 13baerlem3 40226 . . 3 (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)}))))
7473fveq2d 6850 . 2 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ βˆ’ 𝑍)})) = (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ π‘Œ)})(LSSumβ€˜π‘ˆ)(π‘β€˜{(𝑋 βˆ’ 𝑍)})))))
75 eqid 2733 . . 3 (0gβ€˜πΆ) = (0gβ€˜πΆ)
761, 34, 5lcdlvec 40104 . . 3 (πœ‘ β†’ 𝐢 ∈ LVec)
771, 2, 3, 9, 10, 34, 41, 43, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 40184 . . 3 (πœ‘ β†’ Β¬ 𝐹 ∈ (π½β€˜{𝐺, 𝐸}))
781, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 40183 . . 3 (πœ‘ β†’ (π½β€˜{𝐺}) β‰  (π½β€˜{𝐸}))
791, 2, 3, 9, 10, 34, 41, 43, 5, 52, 55, 40, 75, 7mapdn0 40182 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {(0gβ€˜πΆ)}))
801, 2, 3, 9, 10, 34, 41, 43, 5, 60, 63, 40, 75, 13mapdn0 40182 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {(0gβ€˜πΆ)}))
8141, 42, 75, 35, 43, 76, 44, 77, 78, 79, 80baerlem3 40226 . 2 (πœ‘ β†’ (π½β€˜{(𝐺𝑅𝐸)}) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅𝐺)})(LSSumβ€˜πΆ)(π½β€˜{(𝐹𝑅𝐸)}))))
8272, 74, 813eqtr4d 2783 1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ βˆ’ 𝑍)})) = (π½β€˜{(𝐺𝑅𝐸)}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  Vcvv 3447   βˆ– cdif 3911   ∩ cin 3913  ifcif 4490  {csn 4590  {cpr 4592  βŸ¨cotp 4598   ↦ cmpt 5192  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  Basecbs 17091  0gc0g 17329  -gcsg 18758  LSSumclsm 19424  LModclmod 20365  LSubSpclss 20436  LSpanclspn 20476  HLchlt 37862  LHypclh 38497  DVecHcdvh 39591  LCDualclcd 40099  mapdcmpd 40137
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-riotaBAD 37465
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-undef 8208  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-0g 17331  df-mre 17474  df-mrc 17475  df-acs 17477  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-cntz 19105  df-oppg 19132  df-lsm 19426  df-cmn 19572  df-abl 19573  df-mgp 19905  df-ur 19922  df-ring 19974  df-oppr 20057  df-dvdsr 20078  df-unit 20079  df-invr 20109  df-dvr 20120  df-drng 20221  df-lmod 20367  df-lss 20437  df-lsp 20477  df-lvec 20608  df-lsatoms 37488  df-lshyp 37489  df-lcv 37531  df-lfl 37570  df-lkr 37598  df-ldual 37636  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-llines 38011  df-lplanes 38012  df-lvols 38013  df-lines 38014  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-tgrp 39256  df-tendo 39268  df-edring 39270  df-dveca 39516  df-disoa 39542  df-dvech 39592  df-dib 39652  df-dic 39686  df-dih 39742  df-doch 39861  df-djh 39908  df-lcdual 40100  df-mapd 40138
This theorem is referenced by:  mapdheq4  40245
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