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Theorem mapdh6lem2N 40605
Description: Lemma for mapdh6N 40618. Part (6) in [Baer] p. 47, lines 20-22. (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
mapdh6lem2N (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (π½β€˜{(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 mapdh6lem2N
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 39981 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 mapdhe6.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
87eldifad 3961 . . . . . 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 20588 . . . . . 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 mapdhe6.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1413eldifad 3961 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
159, 4, 10lspsncl 20588 . . . . . 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 20694 . . . . 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 3961 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
22 mapdh.p . . . . . . . . 9 + = (+gβ€˜π‘ˆ)
239, 22lmodvacl 20486 . . . . . . . 8 ((π‘ˆ ∈ LMod ∧ π‘Œ ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) β†’ (π‘Œ + 𝑍) ∈ 𝑉)
246, 8, 14, 23syl3anc 1372 . . . . . . 7 (πœ‘ β†’ (π‘Œ + 𝑍) ∈ 𝑉)
25 mapdh.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
269, 25lmodvsubcl 20517 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (π‘Œ + 𝑍) ∈ 𝑉) β†’ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉)
276, 21, 24, 26syl3anc 1372 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉)
289, 4, 10lspsncl 20588 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) ∈ (LSubSpβ€˜π‘ˆ))
296, 27, 28syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) ∈ (LSubSpβ€˜π‘ˆ))
309, 4, 10lspsncl 20588 . . . . . 6 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘ˆ))
316, 21, 30syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘ˆ))
324, 17lsmcl 20694 . . . . 5 ((π‘ˆ ∈ LMod ∧ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) ∈ (LSubSpβ€˜π‘ˆ) ∧ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘ˆ)) β†’ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘ˆ))
336, 29, 31, 32syl3anc 1372 . . . 4 (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})) ∈ (LSubSpβ€˜π‘ˆ))
341, 2, 3, 4, 5, 19, 33mapdin 40533 . . 3 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})))) = ((π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})))))
35 mapdh.c . . . . . 6 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
36 eqid 2733 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 40535 . . . . 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}))))
38 mapdh6.fg . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
39 mapdh.q . . . . . . . . 9 𝑄 = (0gβ€˜πΆ)
40 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)})))))
41 mapdhc.o . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
42 mapdh.d . . . . . . . . 9 𝐷 = (Baseβ€˜πΆ)
43 mapdh.r . . . . . . . . 9 𝑅 = (-gβ€˜πΆ)
44 mapdh.j . . . . . . . . 9 𝐽 = (LSpanβ€˜πΆ)
45 mapdhc.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ 𝐷)
46 mapdh.mn . . . . . . . . 9 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐹}))
471, 3, 5dvhlvec 39980 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LVec)
48 mapdh6.yz . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
49 mapdhe6.xn . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
509, 41, 10, 47, 8, 13, 21, 48, 49lspindp2 20748 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
5150simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
5239, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 8, 51mapdhcl 40598 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5338, 52eqeltrrd 2835 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ 𝐷)
5439, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 7, 53, 51mapdheq 40599 . . . . . . . 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)}))))
5538, 54mpbid 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)})))
5655simpld 496 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝐺}))
57 mapdh6.fe . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
589, 41, 10, 47, 7, 14, 21, 48, 49lspindp1 20746 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) ∧ Β¬ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
5958simpld 496 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
6039, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 14, 59mapdhcl 40598 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6157, 60eqeltrrd 2835 . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ 𝐷)
6239, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 13, 61, 59mapdheq 40599 . . . . . . . 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)}))))
6357, 62mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (π½β€˜{(𝐹𝑅𝐸)})))
6463simpld 496 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (π½β€˜{𝐸}))
6556, 64oveq12d 7427 . . . . 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})))
6637, 65eqtrd 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})))
671, 2, 3, 4, 17, 35, 36, 5, 29, 31mapdlsm 40535 . . . . 5 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋}))) = ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑋}))))
68 mapdh.a . . . . . . 7 ✚ = (+gβ€˜πΆ)
6939, 40, 1, 2, 3, 9, 25, 41, 10, 35, 42, 43, 44, 5, 45, 46, 20, 22, 68, 7, 13, 49, 48, 38, 57mapdh6lem1N 40604 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
7069, 46oveq12d 7427 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑋}))) = ((π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(π½β€˜{𝐹})))
7167, 70eqtrd 2773 . . . 4 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋}))) = ((π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(π½β€˜{𝐹})))
7266, 71ineq12d 4214 . . 3 (πœ‘ β†’ ((π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) ∩ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})))) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(π½β€˜{𝐹}))))
7334, 72eqtrd 2773 . 2 (πœ‘ β†’ (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})))) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(π½β€˜{𝐹}))))
749, 25, 41, 17, 10, 47, 21, 49, 48, 7, 13, 22baerlem5b 40586 . . 3 (πœ‘ β†’ (π‘β€˜{(π‘Œ + 𝑍)}) = (((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋}))))
7574fveq2d 6896 . 2 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (π‘€β€˜(((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋})))))
761, 35, 5lcdlvec 40462 . . 3 (πœ‘ β†’ 𝐢 ∈ LVec)
771, 2, 3, 9, 10, 35, 42, 44, 5, 45, 46, 21, 8, 53, 56, 14, 61, 64, 49mapdindp 40542 . . 3 (πœ‘ β†’ Β¬ 𝐹 ∈ (π½β€˜{𝐺, 𝐸}))
781, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 8, 14, 61, 64, 48mapdncol 40541 . . 3 (πœ‘ β†’ (π½β€˜{𝐺}) β‰  (π½β€˜{𝐸}))
791, 2, 3, 9, 10, 35, 42, 44, 5, 53, 56, 41, 39, 7mapdn0 40540 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {𝑄}))
801, 2, 3, 9, 10, 35, 42, 44, 5, 61, 64, 41, 39, 13mapdn0 40540 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {𝑄}))
8142, 43, 39, 36, 44, 76, 45, 77, 78, 79, 80, 68baerlem5b 40586 . 2 (πœ‘ β†’ (π½β€˜{(𝐺 ✚ 𝐸)}) = (((π½β€˜{𝐺})(LSSumβ€˜πΆ)(π½β€˜{𝐸})) ∩ ((π½β€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(π½β€˜{𝐹}))))
8273, 75, 813eqtr4d 2783 1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (π½β€˜{(𝐺 ✚ 𝐸)}))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948  ifcif 4529  {csn 4629  {cpr 4631  βŸ¨cotp 4637   ↦ cmpt 5232  β€˜cfv 6544  β„©crio 7364  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  Basecbs 17144  +gcplusg 17197  0gc0g 17385  -gcsg 18821  LSSumclsm 19502  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  HLchlt 38220  LHypclh 38855  DVecHcdvh 39949  LCDualclcd 40457  mapdcmpd 40495
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-riotaBAD 37823
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-undef 8258  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-0g 17387  df-mre 17530  df-mrc 17531  df-acs 17533  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-cntz 19181  df-oppg 19210  df-lsm 19504  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-dvr 20215  df-drng 20359  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lvec 20714  df-lsatoms 37846  df-lshyp 37847  df-lcv 37889  df-lfl 37928  df-lkr 37956  df-ldual 37994  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030  df-tgrp 39614  df-tendo 39626  df-edring 39628  df-dveca 39874  df-disoa 39900  df-dvech 39950  df-dib 40010  df-dic 40044  df-dih 40100  df-doch 40219  df-djh 40266  df-lcdual 40458  df-mapd 40496
This theorem is referenced by:  mapdh6aN  40606
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