mapdh6lem1N - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6lem1N		Structured version Visualization version GIF version

Theorem mapdh6lem1N 42234

Description: Lemma for mapdh6N 42248. 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 2739	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
5		mapdh.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 3, 5	dvhlmod 41611	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
7		mapdhcl.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
8	7	eldifad 3895	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
9		mapdhe6.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3895	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11		mapdh.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
12		mapdh.s	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
13	11, 12	lmodvsubcl 20898	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
14	6, 8, 10, 13	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
15		mapdh.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
16	11, 4, 15	lspsncl 20968	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
17	6, 14, 16	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
18		mapdhe6.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
19	18	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
20	11, 4, 15	lspsncl 20968	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
21	6, 19, 20	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
22		eqid 2739	. . . . . 6 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
23	4, 22	lsmcl 21074	. . . . 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 1379	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∈ (LSubSp‘𝑈))
25	11, 12	lmodvsubcl 20898	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑋 − 𝑍) ∈ 𝑉)
26	6, 8, 19, 25	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑍) ∈ 𝑉)
27	11, 4, 15	lspsncl 20968	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑍) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
28	6, 26, 27	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑍)}) ∈ (LSubSp‘𝑈))
29	11, 4, 15	lspsncl 20968	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
30	6, 10, 29	syl2anc 590	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
31	4, 22	lsmcl 21074	. . . . 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 1379	. . . 4 ⊢ (𝜑 → ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
33	1, 2, 3, 4, 5, 24, 32	mapdin 42163	. . 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 2739	. . . . . 6 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
36	1, 2, 3, 4, 22, 34, 35, 5, 17, 21	mapdlsm 42165	. . . . 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 42165	. . . . 5 ⊢ (𝜑 → (𝑀‘((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))) = ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))))
38	36, 37	ineq12d 4151	. . . 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 41610	. . . . . . . . . . . . 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 21129	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
52	51	simpld 495	. . . . . . . . . . 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 42228	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
54	39, 53	eqeltrrd 2840	. . . . . . . . 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 42229	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
56	39, 55	mpbid 233	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
57	56	simprd 496	. . . . . 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 21127	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
60	59	simpld 495	. . . . . . . . . . 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 42228	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
62	58, 61	eqeltrrd 2840	. . . . . . . . 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 42229	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))))
64	58, 63	mpbid 233	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))
65	64	simpld 495	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
66	57, 65	oveq12d 7375	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})))
67	64	simprd 496	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))
68	56	simpld 495	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
69	67, 68	oveq12d 7375	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑍)}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑌}))) = ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺})))
70	66, 69	ineq12d 4151	. . . 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 2774	. . 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 2774	. 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 42215	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)})(LSSum‘𝑈)(𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)})(LSSum‘𝑈)(𝑁‘{𝑌}))))
75	74	fveq2d 6832	. 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 42092	. . 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 42172	. . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))
78	1, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 10, 19, 62, 65, 49	mapdncol 42171	. . 3 ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝐸}))
79	1, 2, 3, 11, 15, 34, 43, 45, 5, 54, 68, 42, 40, 9	mapdn0 42170	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 ∖ {𝑄}))
80	1, 2, 3, 11, 15, 34, 43, 45, 5, 62, 65, 42, 40, 18	mapdn0 42170	. . 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 42215	. 2 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))}) = (((𝐽‘{(𝐹𝑅𝐺)})(LSSum‘𝐶)(𝐽‘{𝐸})) ∩ ((𝐽‘{(𝐹𝑅𝐸)})(LSSum‘𝐶)(𝐽‘{𝐺}))))
83	72, 75, 82	3eqtr4d 2784	1 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 Vcvv 3431 ∖ cdif 3880 ∩ cin 3882 ifcif 4455 {csn 4556 {cpr 4558 ⟨cotp 4564 ↦ cmpt 5154 ‘cfv 6486 ℩crio 7313 (class class class)co 7357 1^st c1st 7930 2^nd c2nd 7931 Basecbs 17171 +_gcplusg 17212 0_gc0g 17394 -_gcsg 18903 LSSumclsm 19601 LModclmod 20851 LSubSpclss 20922 LSpanclspn 20962 HLchlt 39851 LHypclh 40485 DVecHcdvh 41579 LCDualclcd 42087 mapdcmpd 42125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39454

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-ot 4565 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-tpos 8167 df-undef 8214 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-n0 12430 df-z 12517 df-uz 12781 df-fz 13454 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-sca 17228 df-vsca 17229 df-0g 17396 df-mre 17540 df-mrc 17541 df-acs 17543 df-proset 18252 df-poset 18271 df-plt 18286 df-lub 18302 df-glb 18303 df-join 18304 df-meet 18305 df-p0 18381 df-p1 18382 df-lat 18390 df-clat 18457 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-grp 18904 df-minusg 18905 df-sbg 18906 df-subg 19091 df-cntz 19284 df-oppg 19313 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-oppr 20309 df-dvdsr 20329 df-unit 20330 df-invr 20360 df-dvr 20373 df-nzr 20486 df-rlreg 20667 df-domn 20668 df-drng 20704 df-lmod 20853 df-lss 20923 df-lsp 20963 df-lvec 21094 df-lsatoms 39477 df-lshyp 39478 df-lcv 39520 df-lfl 39559 df-lkr 39587 df-ldual 39625 df-oposet 39677 df-ol 39679 df-oml 39680 df-covers 39767 df-ats 39768 df-atl 39799 df-cvlat 39823 df-hlat 39852 df-llines 39999 df-lplanes 40000 df-lvols 40001 df-lines 40002 df-psubsp 40004 df-pmap 40005 df-padd 40297 df-lhyp 40489 df-laut 40490 df-ldil 40605 df-ltrn 40606 df-trl 40660 df-tgrp 41244 df-tendo 41256 df-edring 41258 df-dveca 41504 df-disoa 41530 df-dvech 41580 df-dib 41640 df-dic 41674 df-dih 41730 df-doch 41849 df-djh 41896 df-lcdual 42088 df-mapd 42126

This theorem is referenced by: mapdh6lem2N 42235 mapdh6aN 42236

W3C validator