mapdh6lem1N - Mathbox for Norm Megill

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Theorem mapdh6lem1N 39029

Description: Lemma for mapdh6N 39043. 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 = (0_g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh.v	⊢ 𝑉 = (Base‘𝑈)
mapdh.s	⊢ − = (-_g‘𝑈)
mapdhc.o	⊢ 0 = (0_g‘𝑈)
mapdh.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh.d	⊢ 𝐷 = (Base‘𝐶)
mapdh.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdhc.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdhcl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh.p	⊢ + = (+_g‘𝑈)
mapdh.a	⊢ ✚ = (+_g‘𝐶)
mapdhe6.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdhe6.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdhe6.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh6.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
mapdh6.fg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh6.fe	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)

**Assertion**
Ref	Expression
mapdh6lem1N	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Distinct variable groups: 𝑥,𝐷,ℎ ℎ,𝐹,𝑥 𝑥,𝐽 𝑥,𝑀 𝑥,𝑁 𝑥, 0 𝑥,𝑄 𝑥,𝑅 𝑥, − ℎ,𝑋,𝑥 ℎ,𝑌,𝑥 𝜑,ℎ 0 ,ℎ 𝐶,ℎ 𝐷,ℎ ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ 𝑈,ℎ − ,ℎ ℎ,𝐺,𝑥 ℎ,𝐸 ℎ,𝑍,𝑥 ✚ ,ℎ ℎ,𝐼 + ,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) ✚ (𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐸(𝑥) 𝐻(𝑥,ℎ) 𝐼(𝑥) 𝐾(𝑥,ℎ) 𝑉(𝑥,ℎ) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdh6lem1N**
Step	Hyp	Ref	Expression
1		mapdh.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdh.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		eqid 2798	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdh.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 38406	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		mapdhcl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3893	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		mapdhe6.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3893	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11		mapdh.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
12		mapdh.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
13	11, 12	lmodvsubcl 19672	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
14	6, 8, 10, 13	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
15		mapdh.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
16	11, 4, 15	lspsncl 19742	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
17	6, 14, 16	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
18		mapdhe6.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3893	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
20	11, 4, 15	lspsncl 19742	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
21	6, 19, 20	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
22		eqid 2798	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
23	4, 22	lsmcl 19848	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
24	6, 17, 21, 23	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
25	11, 12	lmodvsubcl 19672	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑋 − 𝑍) ∈ 𝑉)
26	6, 8, 19, 25	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝑉)
27	11, 4, 15	lspsncl 19742	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑍) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
28	6, 26, 27	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
29	11, 4, 15	lspsncl 19742	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
30	6, 10, 29	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
31	4, 22	lsmcl 19848	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
32	6, 28, 30, 31	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
33	1, 2, 3, 4, 5, 24, 32	mapdin 38958	. . 3 ⊢ (𝜑 → (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))))
34		mapdh.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
35		eqid 2798	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
36	1, 2, 3, 4, 22, 34, 35, 5, 17, 21	mapdlsm 38960	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))))
37	1, 2, 3, 4, 22, 34, 35, 5, 28, 30	mapdlsm 38960	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))))
38	36, 37	ineq12d 4140	. . . 4 ⊢ (𝜑 → ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) ∩ ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌})))))
39		mapdh6.fg	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
40		mapdh.q	. . . . . . . . 9 ⊢ 𝑄 = (0_g‘𝐶)
41		mapdh.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
42		mapdhc.o	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
43		mapdh.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐶)
44		mapdh.r	. . . . . . . . 9 ⊢ 𝑅 = (-_g‘𝐶)
45		mapdh.j	. . . . . . . . 9 ⊢ 𝐽 = (LSpan‘𝐶)
46		mapdhc.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
47		mapdh.mn	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
48	1, 3, 5	dvhlvec 38405	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ LVec)
49		mapdh6.yz	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
50		mapdhe6.xn	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
51	11, 42, 15, 48, 10, 18, 8, 49, 50	lspindp2 19900	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
52	51	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
53	40, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 10, 52	mapdhcl 39023	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
54	39, 53	eqeltrrd 2891	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
55	40, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 9, 54, 52	mapdheq 39024	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
56	39, 55	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
57	56	simprd 499	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))
58		mapdh6.fe	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
59	11, 42, 15, 48, 9, 19, 8, 49, 50	lspindp1 19898	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
60	59	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
61	40, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 19, 60	mapdhcl 39023	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
62	58, 61	eqeltrrd 2891	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
63	40, 41, 1, 2, 3, 11, 12, 42, 15, 34, 43, 44, 45, 5, 46, 47, 7, 18, 62, 60	mapdheq 39024	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))))
64	58, 63	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))
65	64	simpld 498	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
66	57, 65	oveq12d 7153	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})))
67	64	simprd 499	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))
68	56	simpld 498	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
69	67, 68	oveq12d 7153	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺})))
70	66, 69	ineq12d 4140	. . . 4 ⊢ (𝜑 → (((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) ∩ ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌})))) = (((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})) ∩ ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺}))))
71	38, 70	eqtrd 2833	. . 3 ⊢ (𝜑 → ((𝑀‘((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍}))) ∩ (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})) ∩ ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺}))))
72	33, 71	eqtrd 2833	. 2 ⊢ (𝜑 → (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))) = (((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})) ∩ ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺}))))
73		mapdh.p	. . . 4 ⊢ + = (+_g‘𝑈)
74	11, 12, 42, 22, 15, 48, 8, 50, 49, 9, 18, 73	baerlem5a 39010	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))))
75	74	fveq2d 6649	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝑀‘(((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})))))
76	1, 34, 5	lcdlvec 38887	. . 3 ⊢ (𝜑 → 𝐶 ∈ LVec)
77	1, 2, 3, 11, 15, 34, 43, 45, 5, 46, 47, 8, 10, 54, 68, 19, 62, 65, 50	mapdindp 38967	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))
78	1, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 10, 19, 62, 65, 49	mapdncol 38966	. . 3 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝐸}))
79	1, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 42, 40, 9	mapdn0 38965	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 11, 15, 34, 43, 45, 5, 62, 65, 42, 40, 18	mapdn0 38965	. . 3 ⊢ (𝜑 → 𝐸 ∈ (𝐷 ∖ {𝑄}))
81		mapdh.a	. . 3 ⊢ ✚ = (+_g‘𝐶)
82	43, 44, 40, 35, 45, 76, 46, 77, 78, 79, 80, 81	baerlem5a 39010	. 2 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})) ∩ ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺}))))
83	72, 75, 82	3eqtr4d 2843	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ifcif 4425 {csn 4525 {cpr 4527 ⟨cotp 4533 ↦ cmpt 5110 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 Basecbs 16475 +_gcplusg 16557 0_gc0g 16705 -_gcsg 18097 LSSumclsm 18751 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 mapdcmpd 38920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921

This theorem is referenced by: mapdh6lem2N 39030 mapdh6aN 39031

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