Theorem lcfrlem23 42064

Description: Lemma for lcfr 42084. 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 6837	. . . . . 6 ⊢ ( ⊥ ‘𝐵) = ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
3		lcfrlem17.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2740	. . . . . . . 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 2740	. . . . . . . 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 3902	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3902	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	3, 5, 6, 10, 4, 9, 12, 14	dihprrn 41925	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
16	3, 5, 9	dvhlmod 41609	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑈)
18	6, 17	lmodvacl 20872	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
19	16, 12, 14, 18	syl3anc 1379	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
20	19	snssd 4725	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
21	3, 4, 5, 6, 7	dochcl 41852	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
22	9, 20, 21	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	3, 4, 5, 6, 7, 8, 9, 15, 22	dochdmm1 41909	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))))
24	3, 5, 7, 6, 10, 9, 19	dochocsn 41880	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
25	24	oveq2d 7379	. . . . . . 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 4759	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
28	12, 14, 27	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
29	6, 10	lspssv 20980	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
30	16, 28, 29	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
31	3, 4, 5, 6, 7	dochcl 41852	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	9, 30, 31	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	3, 5, 6, 26, 10, 4, 8, 9, 32, 19	dihjat1 41928	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
34	23, 25, 33	3eqtrd 2779	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
35	2, 34	eqtrid 2787	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝐵) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
36	35	ineq2d 4156	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))))
37		eqid 2740	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
38	37	lsssssubg 20955	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
39	16, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
40	6, 37, 10	lspsncl 20974	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
41	16, 12, 40	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
42	6, 37, 10	lspsncl 20974	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
43	16, 14, 42	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	37, 26	lsmcl 21080	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
45	16, 41, 43, 44	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
46	39, 45	sseldd 3923	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈))
47	3, 5, 6, 37, 7	dochlss 41853	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
48	9, 30, 47	syl2anc 590	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
49	39, 48	sseldd 3923	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
50	6, 37, 10	lspsncl 20974	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 + 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
51	16, 19, 50	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
52	39, 51	sseldd 3923	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈))
53	6, 17, 10, 26	lspsntri 21094	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
54	16, 12, 14, 53	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
55	26	lsmmod2 19649	. . . . 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 1381	. . . 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 21086	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
58	57	ineq1d 4155	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))))
59	6, 37, 10, 16, 12, 14	lspprcl 20975	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
60		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
61	3, 5, 37, 60, 7	dochnoncon 41890	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
62	9, 59, 61	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
63	58, 62	eqtr3d 2777	. . . . . 6 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
64	63	oveq1d 7378	. . . . 5 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})))
65	60, 26	lsm02 19645	. . . . . 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 2775	. . . 4 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
68	36, 56, 67	3eqtrd 2779	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (𝑁‘{(𝑋 + 𝑌)}))
69	68	fveq2d 6838	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})))
70	3, 5, 6, 26, 10, 4, 9, 12, 14	dihsmsnrn 41934	. . . 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 42063	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
74	6, 71, 16, 73	lsatssv 39497	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
75	3, 4, 5, 6, 7	dochcl 41852	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ⊆ 𝑉) → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	9, 74, 75	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	3, 4, 5, 6, 7, 8, 9, 70, 76	dochdmm1 41909	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))))
78	57	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))))
79	3, 5, 7, 6, 10, 9, 28	dochocsp 41878	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘{𝑋, 𝑌}))
80	78, 79	eqtr3d 2777	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘{𝑋, 𝑌}))
81	3, 5, 4, 71	dih1dimat 41829	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
82	9, 73, 81	syl2anc 590	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
83	3, 4, 7	dochoc 41866	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
84	9, 82, 83	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
85	80, 84	oveq12d 7381	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵))
86	3, 4, 5, 6, 7	dochcl 41852	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
87	9, 28, 86	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
88	3, 4, 8, 5, 26, 71, 9, 87, 73	dihjat2 41930	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
89	77, 85, 88	3eqtrd 2779	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
90	3, 5, 7, 6, 10, 9, 20	dochocsp 41878	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})) = ( ⊥ ‘{(𝑋 + 𝑌)}))
91	69, 89, 90	3eqtr3d 2783	1 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  {csn 4562  {cpr 4564  ran crn 5626  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  SubGrpcsubg 19094  LSSumclsm 19607  LModclmod 20857  LSubSpclss 20928  LSpanclspn 20968  LSAtomsclsa 39473  HLchlt 39849  LHypclh 40483  DVecHcdvh 41577  DIsoHcdih 41727  ocHcoch 41846  joinHcdjh 41893

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-riotaBAD 39452

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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-undef 8220  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-0g 17402  df-mre 17546  df-mrc 17547  df-acs 17549  df-proset 18258  df-poset 18277  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18396  df-clat 18463  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-submnd 18750  df-grp 18910  df-minusg 18911  df-sbg 18912  df-subg 19097  df-cntz 19290  df-oppg 19319  df-lsm 19609  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20315  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-drng 20710  df-lmod 20859  df-lss 20929  df-lsp 20969  df-lvec 21100  df-lsatoms 39475  df-lshyp 39476  df-lcv 39518  df-oposet 39675  df-ol 39677  df-oml 39678  df-covers 39765  df-ats 39766  df-atl 39797  df-cvlat 39821  df-hlat 39850  df-llines 39997  df-lplanes 39998  df-lvols 39999  df-lines 40000  df-psubsp 40002  df-pmap 40003  df-padd 40295  df-lhyp 40487  df-laut 40488  df-ldil 40603  df-ltrn 40604  df-trl 40658  df-tgrp 41242  df-tendo 41254  df-edring 41256  df-dveca 41502  df-disoa 41528  df-dvech 41578  df-dib 41638  df-dic 41672  df-dih 41728  df-doch 41847  df-djh 41894

This theorem is referenced by:  lcfrlem25  42066  lcfrlem35  42076