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Theorem hdmap1l6lem2 40317
Description: Lemma for hdmap1l6 40330. 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 2733 . . . 4 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
5 hdmap1l6.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
61, 3, 5dvhlmod 39619 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ LMod)
7 hdmap1l6e.y . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
87eldifad 3923 . . . . . 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 20453 . . . . . 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 hdmap1l6e.z . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
1413eldifad 3923 . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑉)
159, 4, 10lspsncl 20453 . . . . . 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 20559 . . . . 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 hdmap1l6cl.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2120eldifad 3923 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
22 hdmap1l6.p . . . . . . . . 9 + = (+gβ€˜π‘ˆ)
239, 22lmodvacl 20351 . . . . . . . 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 hdmap1l6.s . . . . . . . 8 βˆ’ = (-gβ€˜π‘ˆ)
269, 25lmodvsubcl 20382 . . . . . . 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 20453 . . . . . 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 20453 . . . . . 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 20559 . . . . 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 40171 . . 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 2733 . . . . . 6 (LSSumβ€˜πΆ) = (LSSumβ€˜πΆ)
371, 2, 3, 4, 17, 35, 36, 5, 12, 16mapdlsm 40173 . . . . 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 39618 . . . . . . . . . . . . 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 20612 . . . . . . . . . . . 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}))
511, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 50, 20, 8hdmap1cl 40313 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
5238, 51eqeltrrd 2835 . . . . . . . . 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 40310 . . . . . . . 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 496 . . . . . 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 20610 . . . . . . . . . . . 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}))
591, 3, 9, 39, 10, 35, 40, 42, 2, 43, 5, 44, 45, 58, 20, 14hdmap1cl 40313 . . . . . . . . . 10 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) ∈ 𝐷)
6056, 59eqeltrrd 2835 . . . . . . . . 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 40310 . . . . . . . 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 496 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑍})) = (πΏβ€˜{𝐸}))
6455, 63oveq12d 7376 . . . . 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 2773 . . . 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 40173 . . . . 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 40316 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ (π‘Œ + 𝑍))})) = (πΏβ€˜{(𝐹𝑅(𝐺 ✚ 𝐸))}))
7069, 45oveq12d 7376 . . . . 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 2773 . . . 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 4174 . . 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 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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 40224 . . 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 6847 . 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 40100 . . 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 40180 . . 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 40179 . . 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 40178 . . 3 (πœ‘ β†’ 𝐺 ∈ (𝐷 βˆ– {𝑄}))
801, 2, 3, 9, 10, 35, 40, 42, 5, 60, 63, 39, 68, 13mapdn0 40178 . . 3 (πœ‘ β†’ 𝐸 ∈ (𝐷 βˆ– {𝑄}))
8140, 41, 68, 36, 42, 76, 44, 77, 78, 79, 80, 67baerlem5b 40224 . 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 2783 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   βˆ– cdif 3908   ∩ cin 3910  {csn 4587  {cpr 4589  βŸ¨cotp 4595  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  -gcsg 18755  LSSumclsm 19421  LModclmod 20336  LSubSpclss 20407  LSpanclspn 20447  HLchlt 37858  LHypclh 38493  DVecHcdvh 39587  LCDualclcd 40095  mapdcmpd 40133  HDMap1chdma1 40300
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-riotaBAD 37461
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-undef 8205  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-0g 17328  df-mre 17471  df-mrc 17472  df-acs 17474  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-cntz 19102  df-oppg 19129  df-lsm 19423  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-dvr 20117  df-drng 20199  df-lmod 20338  df-lss 20408  df-lsp 20448  df-lvec 20579  df-lsatoms 37484  df-lshyp 37485  df-lcv 37527  df-lfl 37566  df-lkr 37594  df-ldual 37632  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859  df-llines 38007  df-lplanes 38008  df-lvols 38009  df-lines 38010  df-psubsp 38012  df-pmap 38013  df-padd 38305  df-lhyp 38497  df-laut 38498  df-ldil 38613  df-ltrn 38614  df-trl 38668  df-tgrp 39252  df-tendo 39264  df-edring 39266  df-dveca 39512  df-disoa 39538  df-dvech 39588  df-dib 39648  df-dic 39682  df-dih 39738  df-doch 39857  df-djh 39904  df-lcdual 40096  df-mapd 40134  df-hdmap1 40302
This theorem is referenced by:  hdmap1l6a  40318
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