Theorem lcfrlem23 39506

Description: Lemma for lcfr 39526. 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 6759	. . . . . 6 ⊢ ( ⊥ ‘𝐵) = ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
3		lcfrlem17.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2738	. . . . . . . 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 2738	. . . . . . . 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 3895	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3895	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	3, 5, 6, 10, 4, 9, 12, 14	dihprrn 39367	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
16	3, 5, 9	dvhlmod 39051	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑈)
18	6, 17	lmodvacl 20052	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
19	16, 12, 14, 18	syl3anc 1369	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
20	19	snssd 4739	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
21	3, 4, 5, 6, 7	dochcl 39294	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
22	9, 20, 21	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	3, 4, 5, 6, 7, 8, 9, 15, 22	dochdmm1 39351	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))))
24	3, 5, 7, 6, 10, 9, 19	dochocsn 39322	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
25	24	oveq2d 7271	. . . . . . 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 4751	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
28	12, 14, 27	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
29	6, 10	lspssv 20160	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ {𝑋, 𝑌} ⊆ 𝑉) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
30	16, 28, 29	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉)
31	3, 4, 5, 6, 7	dochcl 39294	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
32	9, 30, 31	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	3, 5, 6, 26, 10, 4, 8, 9, 32, 19	dihjat1 39370	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
34	23, 25, 33	3eqtrd 2782	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
35	2, 34	syl5eq 2791	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝐵) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
36	35	ineq2d 4143	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))))
37		eqid 2738	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
38	37	lsssssubg 20135	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
39	16, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
40	6, 37, 10	lspsncl 20154	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
41	16, 12, 40	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
42	6, 37, 10	lspsncl 20154	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
43	16, 14, 42	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
44	37, 26	lsmcl 20260	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
45	16, 41, 43, 44	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
46	39, 45	sseldd 3918	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈))
47	3, 5, 6, 37, 7	dochlss 39295	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ⊆ 𝑉) → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
48	9, 30, 47	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (LSubSp‘𝑈))
49	39, 48	sseldd 3918	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
50	6, 37, 10	lspsncl 20154	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 + 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
51	16, 19, 50	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
52	39, 51	sseldd 3918	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈))
53	6, 17, 10, 26	lspsntri 20274	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
54	16, 12, 14, 53	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
55	26	lsmmod2 19197	. . . . 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 1371	. . . 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 20266	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
58	57	ineq1d 4142	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))))
59	6, 37, 10, 16, 12, 14	lspprcl 20155	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
60		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
61	3, 5, 37, 60, 7	dochnoncon 39332	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
62	9, 59, 61	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
63	58, 62	eqtr3d 2780	. . . . . 6 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
64	63	oveq1d 7270	. . . . 5 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})))
65	60, 26	lsm02 19193	. . . . . 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 2778	. . . 4 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
68	36, 56, 67	3eqtrd 2782	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (𝑁‘{(𝑋 + 𝑌)}))
69	68	fveq2d 6760	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})))
70	3, 5, 6, 26, 10, 4, 9, 12, 14	dihsmsnrn 39376	. . . 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 39505	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
74	6, 71, 16, 73	lsatssv 36939	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
75	3, 4, 5, 6, 7	dochcl 39294	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ⊆ 𝑉) → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
76	9, 74, 75	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘𝐵) ∈ ran ((DIsoH‘𝐾)‘𝑊))
77	3, 4, 5, 6, 7, 8, 9, 70, 76	dochdmm1 39351	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))))
78	57	fveq2d 6760	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))))
79	3, 5, 7, 6, 10, 9, 28	dochocsp 39320	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘{𝑋, 𝑌}))
80	78, 79	eqtr3d 2780	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘{𝑋, 𝑌}))
81	3, 5, 4, 71	dih1dimat 39271	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
82	9, 73, 81	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊))
83	3, 4, 7	dochoc 39308	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐵 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
84	9, 82, 83	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝐵)
85	80, 84	oveq12d 7273	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵))
86	3, 4, 5, 6, 7	dochcl 39294	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
87	9, 28, 86	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
88	3, 4, 8, 5, 26, 71, 9, 87, 73	dihjat2 39372	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
89	77, 85, 88	3eqtrd 2782	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
90	3, 5, 7, 6, 10, 9, 20	dochocsp 39320	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})) = ( ⊥ ‘{(𝑋 + 𝑌)}))
91	69, 89, 90	3eqtr3d 2786	1 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  {csn 4558  {cpr 4560  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  SubGrpcsubg 18664  LSSumclsm 19154  LModclmod 20038  LSubSpclss 20108  LSpanclspn 20148  LSAtomsclsa 36915  HLchlt 37291  LHypclh 37925  DVecHcdvh 39019  DIsoHcdih 39169  ocHcoch 39288  joinHcdjh 39335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-undef 8060  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-0g 17069  df-mre 17212  df-mrc 17213  df-acs 17215  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-cntz 18838  df-oppg 18865  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lvec 20280  df-lsatoms 36917  df-lshyp 36918  df-lcv 36960  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tgrp 38684  df-tendo 38696  df-edring 38698  df-dveca 38944  df-disoa 38970  df-dvech 39020  df-dib 39080  df-dic 39114  df-dih 39170  df-doch 39289  df-djh 39336

This theorem is referenced by:  lcfrlem25  39508  lcfrlem35  39518