Theorem lcfrlem23 42153

Description: Lemma for lcfr 42173. 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 6866	. . . . . 6 ⊢ ( ⊥ ‘𝐵) = ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
3		lcfrlem17.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2761	. . . . . . . 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 2761	. . . . . . . 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 3916	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3916	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	3, 5, 6, 10, 4, 9, 12, 14	dihprrn 42014	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
16	3, 5, 9	dvhlmod 41698	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑈)
18	6, 17	lmodvacl 20922	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
19	16, 12, 14, 18	syl3anc 1389	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
20	19	snssd 4744	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
21	3, 4, 5, 6, 7	dochcl 41941	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
22	9, 20, 21	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	3, 4, 5, 6, 7, 8, 9, 15, 22	dochdmm1 41998	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))))
24	3, 5, 7, 6, 10, 9, 19	dochocsn 41969	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
25	24	oveq2d 7408	. . . . . . 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 4778	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
28	12, 14, 27	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
29	6, 10	lspssv 21030	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
30	16, 28, 29	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
31	3, 4, 5, 6, 7	dochcl 41941	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	9, 30, 31	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	3, 5, 6, 26, 10, 4, 8, 9, 32, 19	dihjat1 42017	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
34	23, 25, 33	3eqtrd 2800	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
35	2, 34	eqtrid 2808	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝐵) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
36	35	ineq2d 4172	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))))
37		eqid 2761	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
38	37	lsssssubg 21005	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
39	16, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
40	6, 37, 10	lspsncl 21024	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
41	16, 12, 40	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
42	6, 37, 10	lspsncl 21024	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
43	16, 14, 42	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	37, 26	lsmcl 21130	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
45	16, 41, 43, 44	syl3anc 1389	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
46	39, 45	sseldd 3937	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈))
47	3, 5, 6, 37, 7	dochlss 41942	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
48	9, 30, 47	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
49	39, 48	sseldd 3937	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
50	6, 37, 10	lspsncl 21024	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 + 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
51	16, 19, 50	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
52	39, 51	sseldd 3937	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈))
53	6, 17, 10, 26	lspsntri 21144	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
54	16, 12, 14, 53	syl3anc 1389	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
55	26	lsmmod2 19699	. . . . 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 1391	. . . 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 21136	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
58	57	ineq1d 4171	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))))
59	6, 37, 10, 16, 12, 14	lspprcl 21025	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
60		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
61	3, 5, 37, 60, 7	dochnoncon 41979	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
62	9, 59, 61	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
63	58, 62	eqtr3d 2798	. . . . . 6 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
64	63	oveq1d 7407	. . . . 5 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})))
65	60, 26	lsm02 19695	. . . . . 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 2796	. . . 4 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
68	36, 56, 67	3eqtrd 2800	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (𝑁‘{(𝑋 + 𝑌)}))
69	68	fveq2d 6867	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})))
70	3, 5, 6, 26, 10, 4, 9, 12, 14	dihsmsnrn 42023	. . . 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 42152	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
74	6, 71, 16, 73	lsatssv 39586	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
75	3, 4, 5, 6, 7	dochcl 41941	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ⊆ 𝑉) → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	9, 74, 75	syl2anc 593	. . . 4 ⊢ (𝜑 → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	3, 4, 5, 6, 7, 8, 9, 70, 76	dochdmm1 41998	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))))
78	57	fveq2d 6867	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))))
79	3, 5, 7, 6, 10, 9, 28	dochocsp 41967	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘{𝑋, 𝑌}))
80	78, 79	eqtr3d 2798	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘{𝑋, 𝑌}))
81	3, 5, 4, 71	dih1dimat 41918	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
82	9, 73, 81	syl2anc 593	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
83	3, 4, 7	dochoc 41955	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
84	9, 82, 83	syl2anc 593	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
85	80, 84	oveq12d 7410	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵))
86	3, 4, 5, 6, 7	dochcl 41941	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
87	9, 28, 86	syl2anc 593	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
88	3, 4, 8, 5, 26, 71, 9, 87, 73	dihjat2 42019	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
89	77, 85, 88	3eqtrd 2800	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
90	3, 5, 7, 6, 10, 9, 20	dochocsp 41967	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})) = ( ⊥ ‘{(𝑋 + 𝑌)}))
91	69, 89, 90	3eqtr3d 2804	1 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  {csn 4581  {cpr 4583  ran crn 5646  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  SubGrpcsubg 19145  LSSumclsm 19657  LModclmod 20907  LSubSpclss 20978  LSpanclspn 21018  LSAtomsclsa 39562  HLchlt 39938  LHypclh 40572  DVecHcdvh 41666  DIsoHcdih 41816  ocHcoch 41935  joinHcdjh 41982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-riotaBAD 39541

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-tpos 8201  df-undef 8248  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-0g 17453  df-mre 17597  df-mrc 17598  df-acs 17600  df-proset 18309  df-poset 18328  df-plt 18343  df-lub 18359  df-glb 18360  df-join 18361  df-meet 18362  df-p0 18438  df-p1 18439  df-lat 18447  df-clat 18514  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-submnd 18801  df-grp 18961  df-minusg 18962  df-sbg 18963  df-subg 19148  df-cntz 19340  df-oppg 19369  df-lsm 19659  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-ring 20264  df-oppr 20365  df-dvdsr 20385  df-unit 20386  df-invr 20416  df-dvr 20429  df-drng 20760  df-lmod 20909  df-lss 20979  df-lsp 21019  df-lvec 21150  df-lsatoms 39564  df-lshyp 39565  df-lcv 39607  df-oposet 39764  df-ol 39766  df-oml 39767  df-covers 39854  df-ats 39855  df-atl 39886  df-cvlat 39910  df-hlat 39939  df-llines 40086  df-lplanes 40087  df-lvols 40088  df-lines 40089  df-psubsp 40091  df-pmap 40092  df-padd 40384  df-lhyp 40576  df-laut 40577  df-ldil 40692  df-ltrn 40693  df-trl 40747  df-tgrp 41331  df-tendo 41343  df-edring 41345  df-dveca 41591  df-disoa 41617  df-dvech 41667  df-dib 41727  df-dic 41761  df-dih 41817  df-doch 41936  df-djh 41983

This theorem is referenced by:  lcfrlem25  42155  lcfrlem35  42165