Theorem lcfrlem23 40949

Description: Lemma for lcfr 40969. 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 6888	. . . . . 6 ⊢ ( ⊥ ‘𝐵) = ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
3		lcfrlem17.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2726	. . . . . . . 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 2726	. . . . . . . 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 3955	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
13		lcfrlem17.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3955	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	3, 5, 6, 10, 4, 9, 12, 14	dihprrn 40810	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
16	3, 5, 9	dvhlmod 40494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LMod)
17		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑈)
18	6, 17	lmodvacl 20721	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
19	16, 12, 14, 18	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
20	19	snssd 4807	. . . . . . . . 9 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
21	3, 4, 5, 6, 7	dochcl 40737	. . . . . . . . 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 40794	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)}))))
24	3, 5, 7, 6, 10, 9, 19	dochocsn 40765	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
25	24	oveq2d 7421	. . . . . . 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 4819	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
28	12, 14, 27	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
29	6, 10	lspssv 20830	. . . . . . . . . 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 40737	. . . . . . . . 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 40813	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘(𝑁‘{𝑋, 𝑌}))((joinH‘𝐾)‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
34	23, 25, 33	3eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
35	2, 34	eqtrid 2778	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝐵) = (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)})))
36	35	ineq2d 4207	. . . 4 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ (( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ⊕ (𝑁‘{(𝑋 + 𝑌)}))))
37		eqid 2726	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
38	37	lsssssubg 20805	. . . . . . 7 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
39	16, 38	syl 17	. . . . . 6 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
40	6, 37, 10	lspsncl 20824	. . . . . . . 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 20824	. . . . . . . 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 20931	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
45	16, 41, 43, 44	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (LSubSp‘𝑈))
46	39, 45	sseldd 3978	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ (SubGrp‘𝑈))
47	3, 5, 6, 37, 7	dochlss 40738	. . . . . . 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 3978	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) ∈ (SubGrp‘𝑈))
50	6, 37, 10	lspsncl 20824	. . . . . . 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 3978	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ∈ (SubGrp‘𝑈))
53	6, 17, 10, 26	lspsntri 20945	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
54	16, 12, 14, 53	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
55	26	lsmmod2 19596	. . . . 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 1370	. . . 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 20937	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))
58	57	ineq1d 4206	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))))
59	6, 37, 10, 16, 12, 14	lspprcl 20825	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
60		lcfrlem17.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑈)
61	3, 5, 37, 60, 7	dochnoncon 40775	. . . . . . . 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 2768	. . . . . 6 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) = { 0 })
64	63	oveq1d 7420	. . . . 5 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = ({ 0 } ⊕ (𝑁‘{(𝑋 + 𝑌)})))
65	60, 26	lsm02 19592	. . . . . 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 2766	. . . 4 ⊢ (𝜑 → ((((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘(𝑁‘{𝑋, 𝑌}))) ⊕ (𝑁‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
68	36, 56, 67	3eqtrd 2770	. . 3 ⊢ (𝜑 → (((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵)) = (𝑁‘{(𝑋 + 𝑌)}))
69	68	fveq2d 6889	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})))
70	3, 5, 6, 26, 10, 4, 9, 12, 14	dihsmsnrn 40819	. . . 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 40948	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
74	6, 71, 16, 73	lsatssv 38381	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
75	3, 4, 5, 6, 7	dochcl 40737	. . . . 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 40794	. . 3 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))))
78	57	fveq2d 6889	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))))
79	3, 5, 7, 6, 10, 9, 28	dochocsp 40763	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋, 𝑌})) = ( ⊥ ‘{𝑋, 𝑌}))
80	78, 79	eqtr3d 2768	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) = ( ⊥ ‘{𝑋, 𝑌}))
81	3, 5, 4, 71	dih1dimat 40714	. . . . . 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 40751	. . . . 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 7423	. . 3 ⊢ (𝜑 → (( ⊥ ‘((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})))((joinH‘𝐾)‘𝑊)( ⊥ ‘( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵))
86	3, 4, 5, 6, 7	dochcl 40737	. . . . 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 40815	. . 3 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌})((joinH‘𝐾)‘𝑊)𝐵) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
89	77, 85, 88	3eqtrd 2770	. 2 ⊢ (𝜑 → ( ⊥ ‘(((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∩ ( ⊥ ‘𝐵))) = (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵))
90	3, 5, 7, 6, 10, 9, 20	dochocsp 40763	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{(𝑋 + 𝑌)})) = ( ⊥ ‘{(𝑋 + 𝑌)}))
91	69, 89, 90	3eqtr3d 2774	1 ⊢ (𝜑 → (( ⊥ ‘{𝑋, 𝑌}) ⊕ 𝐵) = ( ⊥ ‘{(𝑋 + 𝑌)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2934   ∖ cdif 3940   ∩ cin 3942   ⊆ wss 3943  {csn 4623  {cpr 4625  ran crn 5670  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  +_gcplusg 17206  0_gc0g 17394  SubGrpcsubg 19047  LSSumclsm 19554  LModclmod 20706  LSubSpclss 20778  LSpanclspn 20818  LSAtomsclsa 38357  HLchlt 38733  LHypclh 39368  DVecHcdvh 40462  DIsoHcdih 40612  ocHcoch 40731  joinHcdjh 40778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-riotaBAD 38336

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-undef 8259  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-0g 17396  df-mre 17539  df-mrc 17540  df-acs 17542  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-cntz 19233  df-oppg 19262  df-lsm 19556  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-dvr 20303  df-drng 20589  df-lmod 20708  df-lss 20779  df-lsp 20819  df-lvec 20951  df-lsatoms 38359  df-lshyp 38360  df-lcv 38402  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-llines 38882  df-lplanes 38883  df-lvols 38884  df-lines 38885  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372  df-laut 39373  df-ldil 39488  df-ltrn 39489  df-trl 39543  df-tgrp 40127  df-tendo 40139  df-edring 40141  df-dveca 40387  df-disoa 40413  df-dvech 40463  df-dib 40523  df-dic 40557  df-dih 40613  df-doch 40732  df-djh 40779

This theorem is referenced by:  lcfrlem25  40951  lcfrlem35  40961