Theorem lcfrlem23 37641

Description: Lemma for lcfr 37661. TODO: this proof was built from other proof pieces that may change 𝑁‘{𝑋, 𝑌} into subspace sum and back unnecessarily, or similar things. (Contributed by NM, 1-Mar-2015.)

Hypotheses

Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+g‘𝑈)
lcfrlem17.z	⊢ 0 = (0g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem23.s	⊢ ⊕ = (LSSum‘𝑈)

Assertion

Ref	Expression
lcfrlem23	⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Proof of Theorem lcfrlem23

Step	Hyp	Ref	Expression
1		lcfrlem22.b	. . . . . . 7 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
2	1	fveq2i 6437	. . . . . 6 ⊢ ( ⊥ ‘𝐵) = ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
3		lcfrlem17.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2826	. . . . . . . 8 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
5		lcfrlem17.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcfrlem17.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
7		lcfrlem17.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
8		eqid 2826	. . . . . . . 8 ⊢ ((joinH‘𝐾)‘𝑊) = ((joinH‘𝐾)‘𝑊)
9		lcfrlem17.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		lcfrlem17.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
11		lcfrlem17.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
12	11	eldifad 3811	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3811	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	3, 5, 6, 10, 4, 9, 12, 14	dihprrn 37502	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
16	3, 5, 9	dvhlmod 37186	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑈)
18	6, 17	lmodvacl 19234	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
19	16, 12, 14, 18	syl3anc 1496	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
20	19	snssd 4559	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
21	3, 4, 5, 6, 7	dochcl 37429	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
22	9, 20, 21	syl2anc 581	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	3, 4, 5, 6, 7, 8, 9, 15, 22	dochdmm1 37486	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))))
24	3, 5, 7, 6, 10, 9, 19	dochocsn 37457	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
25	24	oveq2d 6922	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
26		lcfrlem23.s	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑈)
27		prssi 4571	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
28	12, 14, 27	syl2anc 581	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
29	6, 10	lspssv 19343	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
30	16, 28, 29	syl2anc 581	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
31	3, 4, 5, 6, 7	dochcl 37429	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	9, 30, 31	syl2anc 581	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	3, 5, 6, 26, 10, 4, 8, 9, 32, 19	dihjat1 37505	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
34	23, 25, 33	3eqtrd 2866	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
35	2, 34	syl5eq 2874	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝐵) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
36	35	ineq2d 4042	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))))
37		eqid 2826	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
38	37	lsssssubg 19318	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
39	16, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
40	6, 37, 10	lspsncl 19337	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
41	16, 12, 40	syl2anc 581	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
42	6, 37, 10	lspsncl 19337	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
43	16, 14, 42	syl2anc 581	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	37, 26	lsmcl 19443	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
45	16, 41, 43, 44	syl3anc 1496	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
46	39, 45	sseldd 3829	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈))
47	3, 5, 6, 37, 7	dochlss 37430	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
48	9, 30, 47	syl2anc 581	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
49	39, 48	sseldd 3829	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
50	6, 37, 10	lspsncl 19337	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 + 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
51	16, 19, 50	syl2anc 581	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
52	39, 51	sseldd 3829	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈))
53	6, 17, 10, 26	lspsntri 19457	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
54	16, 12, 14, 53	syl3anc 1496	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
55	26	lsmmod2 18441	. . . . 5 ⊢ (((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈) ∧ ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈)) ∧ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))) = ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
56	46, 49, 52, 54, 55	syl31anc 1498	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))) = ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
57	6, 10, 26, 16, 12, 14	lsmpr 19449	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
58	57	ineq1d 4041	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))))
59	6, 37, 10, 16, 12, 14	lspprcl 19338	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
60		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
61	3, 5, 37, 60, 7	dochnoncon 37467	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
62	9, 59, 61	syl2anc 581	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
63	58, 62	eqtr3d 2864	. . . . . 6 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
64	63	oveq1d 6921	. . . . 5 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})))
65	60, 26	lsm02 18437	. . . . . 6 ⊢ ((𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈) → ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
66	52, 65	syl 17	. . . . 5 ⊢ (𝜑 → ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
67	64, 66	eqtrd 2862	. . . 4 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
68	36, 56, 67	3eqtrd 2866	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (𝑁‘{(𝑋 + 𝑌)}))
69	68	fveq2d 6438	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})))
70	3, 5, 6, 26, 10, 4, 9, 12, 14	dihsmsnrn 37511	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
71		lcfrlem17.a	. . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈)
72		lcfrlem17.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
73	3, 7, 5, 6, 17, 60, 10, 71, 9, 11, 13, 72, 1	lcfrlem22 37640	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
74	6, 71, 16, 73	lsatssv 35074	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
75	3, 4, 5, 6, 7	dochcl 37429	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ⊆ 𝑉) → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	9, 74, 75	syl2anc 581	. . . 4 ⊢ (𝜑 → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	3, 4, 5, 6, 7, 8, 9, 70, 76	dochdmm1 37486	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))))
78	57	fveq2d 6438	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))))
79	3, 5, 7, 6, 10, 9, 28	dochocsp 37455	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘{𝑋, 𝑌}))
80	78, 79	eqtr3d 2864	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘{𝑋, 𝑌}))
81	3, 5, 4, 71	dih1dimat 37406	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
82	9, 73, 81	syl2anc 581	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
83	3, 4, 7	dochoc 37443	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
84	9, 82, 83	syl2anc 581	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
85	80, 84	oveq12d 6924	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵))
86	3, 4, 5, 6, 7	dochcl 37429	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
87	9, 28, 86	syl2anc 581	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
88	3, 4, 8, 5, 26, 71, 9, 87, 73	dihjat2 37507	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
89	77, 85, 88	3eqtrd 2866	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
90	3, 5, 7, 6, 10, 9, 20	dochocsp 37455	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})) = ( ⊥ ‘{(𝑋 + 𝑌)}))
91	69, 89, 90	3eqtr3d 2870	1 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ≠ wne 3000   ∖ cdif 3796   ∩ cin 3798   ⊆ wss 3799  {csn 4398  {cpr 4400  ran crn 5344  ‘cfv 6124  (class class class)co 6906  Basecbs 16223  +_gcplusg 16306  0_gc0g 16454  SubGrpcsubg 17940  LSSumclsm 18401  LModclmod 19220  LSubSpclss 19289  LSpanclspn 19331  LSAtomsclsa 35050  HLchlt 35426  LHypclh 36060  DVecHcdvh 37154  DIsoHcdih 37304  ocHcoch 37423  joinHcdjh 37470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330  ax-riotaBAD 35029

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-iin 4744  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-tpos 7618  df-undef 7665  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-struct 16225  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-plusg 16319  df-mulr 16320  df-sca 16322  df-vsca 16323  df-0g 16456  df-mre 16600  df-mrc 16601  df-acs 16603  df-proset 17282  df-poset 17300  df-plt 17312  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-p0 17393  df-p1 17394  df-lat 17400  df-clat 17462  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-submnd 17690  df-grp 17780  df-minusg 17781  df-sbg 17782  df-subg 17943  df-cntz 18101  df-oppg 18127  df-lsm 18403  df-cmn 18549  df-abl 18550  df-mgp 18845  df-ur 18857  df-ring 18904  df-oppr 18978  df-dvdsr 18996  df-unit 18997  df-invr 19027  df-dvr 19038  df-drng 19106  df-lmod 19222  df-lss 19290  df-lsp 19332  df-lvec 19463  df-lsatoms 35052  df-lshyp 35053  df-lcv 35095  df-oposet 35252  df-ol 35254  df-oml 35255  df-covers 35342  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427  df-llines 35574  df-lplanes 35575  df-lvols 35576  df-lines 35577  df-psubsp 35579  df-pmap 35580  df-padd 35872  df-lhyp 36064  df-laut 36065  df-ldil 36180  df-ltrn 36181  df-trl 36235  df-tgrp 36819  df-tendo 36831  df-edring 36833  df-dveca 37079  df-disoa 37105  df-dvech 37155  df-dib 37215  df-dic 37249  df-dih 37305  df-doch 37424  df-djh 37471

This theorem is referenced by:  lcfrlem25  37643  lcfrlem35  37653