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Theorem hdmap1l6lem2 41333
Description: Lemma for hdmap1l6 41346. Part (6) in [Baer] p. 47, lines 20-22. (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
hdmap1l6lem2 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (πΏβ€˜{(𝐺 ✚ 𝐸)}))

Proof of Theorem hdmap1l6lem2
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 40635 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 hdmap1l6e.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
87eldifad 3953 . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑉)
9 hdmap1l6.v . . . . . . 7 𝑉 = (Baseβ€˜π‘ˆ)
10 hdmap1l6.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘ˆ)
119, 4, 10lspsncl 20860 . . . . . 6 ((π‘ˆ ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
126, 8, 11syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘ˆ))
13 hdmap1l6e.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1413eldifad 3953 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
159, 4, 10lspsncl 20860 . . . . . 6 ((π‘ˆ ∈ LMod ∧ 𝑍 ∈ 𝑉) β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
166, 14, 15syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑍}) ∈ (LSubSpβ€˜π‘ˆ))
17 eqid 2725 . . . . . 6 (LSSumβ€˜π‘ˆ) = (LSSumβ€˜π‘ˆ)
184, 17lsmcl 20967 . . . . 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 1368 . . . 4 (πœ‘ β†’ ((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍})) ∈ (LSubSpβ€˜π‘ˆ))
20 hdmap1l6cl.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2120eldifad 3953 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
22 hdmap1l6.p . . . . . . . . 9 + = (+gβ€˜π‘ˆ)
239, 22lmodvacl 20757 . . . . . . . 8 ((π‘ˆ ∈ LMod ∧ π‘Œ ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) β†’ (π‘Œ + 𝑍) ∈ 𝑉)
246, 8, 14, 23syl3anc 1368 . . . . . . 7 (πœ‘ β†’ (π‘Œ + 𝑍) ∈ 𝑉)
25 hdmap1l6.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
269, 25lmodvsubcl 20789 . . . . . . 7 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ (π‘Œ + 𝑍) ∈ 𝑉) β†’ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉)
276, 21, 24, 26syl3anc 1368 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉)
289, 4, 10lspsncl 20860 . . . . . 6 ((π‘ˆ ∈ LMod ∧ (𝑋 βˆ’ (π‘Œ + 𝑍)) ∈ 𝑉) β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) ∈ (LSubSpβ€˜π‘ˆ))
296, 27, 28syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}) ∈ (LSubSpβ€˜π‘ˆ))
309, 4, 10lspsncl 20860 . . . . . 6 ((π‘ˆ ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘ˆ))
316, 21, 30syl2anc 582 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) ∈ (LSubSpβ€˜π‘ˆ))
324, 17lsmcl 20967 . . . . 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 1368 . . . 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 41187 . . 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 hdmap1l6.c . . . . . 6 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
36 eqid 2725 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 41189 . . . . 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 hdmap1l6.fg . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
39 hdmap1l6c.o . . . . . . . . 9 0 = (0gβ€˜π‘ˆ)
40 hdmap1l6.d . . . . . . . . 9 𝐷 = (Baseβ€˜πΆ)
41 hdmap1l6.r . . . . . . . . 9 𝑅 = (-gβ€˜πΆ)
42 hdmap1l6.l . . . . . . . . 9 𝐿 = (LSpanβ€˜πΆ)
43 hdmap1l6.i . . . . . . . . 9 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
44 hdmap1l6.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 ∈ 𝐷)
45 hdmap1l6.mn . . . . . . . . . . 11 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (πΏβ€˜{𝐹}))
461, 3, 5dvhlvec 40634 . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ LVec)
47 hdmap1l6.yz . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
48 hdmap1l6e.xn . . . . . . . . . . . . 13 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
499, 39, 10, 46, 8, 13, 21, 47, 48lspindp2 21022 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ Β¬ 𝑍 ∈ (π‘β€˜{𝑋, π‘Œ})))
5049simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
511, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 50, 20, 8hdmap1cl 41329 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5238, 51eqeltrrd 2826 . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ 𝐷)
531, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 7, 52, 50, 45hdmap1eq 41326 . . . . . . . 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)}))))
5438, 53mpbid 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)})))
5554simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{π‘Œ})) = (πΏβ€˜{𝐺}))
56 hdmap1l6.fe . . . . . . . 8 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐸)
579, 39, 10, 46, 7, 14, 21, 47, 48lspindp1 21020 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}) ∧ Β¬ π‘Œ ∈ (π‘β€˜{𝑋, 𝑍})))
5857simpld 493 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍}))
591, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 58, 20, 14hdmap1cl 41329 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6056, 59eqeltrrd 2826 . . . . . . . . 9 (πœ‘ β†’ 𝐸 ∈ 𝐷)
611, 3, 9, 25, 39, 10, 35, 40, 41, 42, 2, 43, 5, 20, 44, 13, 60, 58, 45hdmap1eq 41326 . . . . . . . 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)}))))
6256, 61mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ 𝑍)})) = (πΏβ€˜{(𝐹𝑅𝐸)})))
6362simpld 493 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}))
6455, 63oveq12d 7431 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{π‘Œ}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑍}))) = ((πΏβ€˜{𝐺})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})))
6537, 64eqtrd 2765 . . . 4 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{π‘Œ})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑍}))) = ((πΏβ€˜{𝐺})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})))
661, 2, 3, 4, 17, 35, 36, 5, 29, 31mapdlsm 41189 . . . . 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}))))
67 hdmap1l6.a . . . . . . 7 ✚ = (+gβ€˜πΆ)
68 hdmap1l6.q . . . . . . 7 𝑄 = (0gβ€˜πΆ)
691, 3, 9, 22, 25, 39, 10, 35, 40, 67, 41, 68, 42, 2, 43, 5, 44, 20, 45, 7, 13, 48, 47, 38, 56hdmap1l6lem1 41332 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
7069, 45oveq12d 7431 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))}))(LSSumβ€˜πΆ)(π‘€β€˜(π‘β€˜{𝑋}))) = ((πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(πΏβ€˜{𝐹})))
7166, 70eqtrd 2765 . . . 4 (πœ‘ β†’ (π‘€β€˜((π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})(LSSumβ€˜π‘ˆ)(π‘β€˜{𝑋}))) = ((πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(πΏβ€˜{𝐹})))
7265, 71ineq12d 4208 . . 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Ώβ€˜{𝐺})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(πΏβ€˜{𝐹}))))
7334, 72eqtrd 2765 . 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Ώβ€˜{𝐺})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(πΏβ€˜{𝐹}))))
749, 25, 39, 17, 10, 46, 21, 48, 47, 7, 13, 22baerlem5b 41240 . . 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 6894 . 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 41116 . . 3 (πœ‘ β†’ 𝐢 ∈ LVec)
771, 2, 3, 9, 10, 35, 40, 42, 5, 44, 45, 21, 8, 52, 55, 14, 60, 63, 48mapdindp 41196 . . 3 (πœ‘ β†’ Β¬ 𝐹 ∈ (πΏβ€˜{𝐺, 𝐸}))
781, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 8, 14, 60, 63, 47mapdncol 41195 . . 3 (πœ‘ β†’ (πΏβ€˜{𝐺}) β‰  (πΏβ€˜{𝐸}))
791, 2, 3, 9, 10, 35, 40, 42, 5, 52, 55, 39, 68, 7mapdn0 41194 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {𝑄}))
801, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13mapdn0 41194 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {𝑄}))
8140, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67baerlem5b 41240 . 2 (πœ‘ β†’ (πΏβ€˜{(𝐺 ✚ 𝐸)}) = (((πΏβ€˜{𝐺})(LSSumβ€˜πΆ)(πΏβ€˜{𝐸})) ∩ ((πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))})(LSSumβ€˜πΆ)(πΏβ€˜{𝐹}))))
8273, 75, 813eqtr4d 2775 1 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(π‘Œ + 𝑍)})) = (πΏβ€˜{(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 3938   ∩ cin 3940  {csn 4625  {cpr 4627  βŸ¨cotp 4633  β€˜cfv 6543  (class class class)co 7413  Basecbs 17174  +gcplusg 17227  0gc0g 17415  -gcsg 18891  LSSumclsm 19588  LModclmod 20742  LSubSpclss 20814  LSpanclspn 20854  HLchlt 38874  LHypclh 39509  DVecHcdvh 40603  LCDualclcd 41111  mapdcmpd 41149  HDMap1chdma1 41316
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-riotaBAD 38477
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-undef 8272  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-n0 12498  df-z 12584  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-0g 17417  df-mre 17560  df-mrc 17561  df-acs 17563  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-p1 18412  df-lat 18418  df-clat 18485  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-submnd 18735  df-grp 18892  df-minusg 18893  df-sbg 18894  df-subg 19077  df-cntz 19267  df-oppg 19296  df-lsm 19590  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-oppr 20272  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-drng 20625  df-lmod 20744  df-lss 20815  df-lsp 20855  df-lvec 20987  df-lsatoms 38500  df-lshyp 38501  df-lcv 38543  df-lfl 38582  df-lkr 38610  df-ldual 38648  df-oposet 38700  df-ol 38702  df-oml 38703  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-llines 39023  df-lplanes 39024  df-lvols 39025  df-lines 39026  df-psubsp 39028  df-pmap 39029  df-padd 39321  df-lhyp 39513  df-laut 39514  df-ldil 39629  df-ltrn 39630  df-trl 39684  df-tgrp 40268  df-tendo 40280  df-edring 40282  df-dveca 40528  df-disoa 40554  df-dvech 40604  df-dib 40664  df-dic 40698  df-dih 40754  df-doch 40873  df-djh 40920  df-lcdual 41112  df-mapd 41150  df-hdmap1 41318
This theorem is referenced by:  hdmap1l6a  41334
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