hdmap11lem2 - Mathbox for Norm Megill

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Theorem hdmap11lem2 41887

Description: Lemma for hdmapadd 41888. (Contributed by NM, 26-May-2015.)

**Hypotheses**
Ref	Expression
hdmap11.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap11.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap11.v	⊢ 𝑉 = (Base‘𝑈)
hdmap11.p	⊢ + = (+_g‘𝑈)
hdmap11.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap11.a	⊢ ✚ = (+_g‘𝐶)
hdmap11.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap11.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap11.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmap11.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
hdmap11.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmap11.o	⊢ 0 = (0_g‘𝑈)
hdmap11.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap11.d	⊢ 𝐷 = (Base‘𝐶)
hdmap11.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmap11.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap11.j	⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmap11.i	⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
hdmap11lem2	⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))

Proof of Theorem hdmap11lem2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmap11.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap11.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmap11.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
4		hdmap11.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
5		hdmap11.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		hdmap11.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		hdmap11.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
8	1, 2, 3, 4, 5, 6, 7	dvh3dim 41491	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
10		eqid 2731	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
11	1, 2, 5	dvhlmod 41155	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
13	3, 10, 4, 11, 6, 7	lspprcl 20912	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
15		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐸 ∈ (𝑁‘{𝑋, 𝑌}))
16	10, 4, 12, 14, 15	ellspsn5 20930	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝑋, 𝑌}))
17	16	ssneld 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝐸})))
18	17	ancld 550	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
19	18	reximdv 3147	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
20	9, 19	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
21		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		eqid 2731	. . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
23		hdmap11.o	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑈)
24		hdmap11.e	. . . . . . . . . 10 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
25	1, 21, 22, 2, 3, 23, 24, 5	dvheveccl 41157	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))
26	25	eldifad 3914	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
27	1, 2, 3, 4, 5, 26, 7	dvh3dim 41491	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}))
28	27	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}))
29		preq1 4686	. . . . . . . . . . . . 13 ⊢ (𝑋 = 0 → {𝑋, 𝑌} = { 0 , 𝑌})
30		prcom 4685	. . . . . . . . . . . . 13 ⊢ { 0 , 𝑌} = {𝑌, 0 }
31	29, 30	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑋 = 0 → {𝑋, 𝑌} = {𝑌, 0 })
32	31	fveq2d 6826	. . . . . . . . . . 11 ⊢ (𝑋 = 0 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 0 }))
33	3, 23, 4, 11, 7	lsppr0 21027	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
34	32, 33	sylan9eqr 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌}))
35	3, 10, 4, 11, 26, 7	lspprcl 20912	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑌}) ∈ (LSubSp‘𝑈))
36	3, 4, 11, 26, 7	lspprid2 20932	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝐸, 𝑌}))
37	10, 4, 11, 35, 36	ellspsn5 20930	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
39	34, 38	eqsstrd 3969	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
40	39	ssneld 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
41	3, 4, 11, 26, 7	lspprid1 20931	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑌}))
42	10, 4, 11, 35, 41	ellspsn5 20930	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑌}))
43	42	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑌}))
44	43	ssneld 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝐸})))
45	40, 44	jcad 512	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
46	45	reximdv 3147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
47	28, 46	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
48	47	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 = 0 ) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
49		hdmap11.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
50	3, 49	lmodvacl 20809	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝐸 + 𝑋) ∈ 𝑉)
51	11, 26, 6, 50	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐸 + 𝑋) ∈ 𝑉)
52	51	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝐸 + 𝑋) ∈ 𝑉)
53	11	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑈 ∈ LMod)
54	13	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
55	3, 4, 11, 6, 7	lspprid1 20931	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑌}))
56	55	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑁‘{𝑋, 𝑌}))
57	26	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝐸 ∈ 𝑉)
58		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌}))
59	3, 49, 10, 53, 54, 56, 57, 58	lssvancl2 20880	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌}))
60	3, 10, 4	lspsncl 20911	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
61	11, 26, 60	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
62	61	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
63	3, 4	lspsnid 20927	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ (𝑁‘{𝐸}))
64	11, 26, 63	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸}))
65	64	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝐸 ∈ (𝑁‘{𝐸}))
66	6	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉)
67	1, 2, 5	dvhlvec 41154	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
68	67	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑈 ∈ LVec)
69		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
70		eldifsn 4738	. . . . . . . 8 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
71	66, 69, 70	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
72	10, 4, 11, 13, 55	ellspsn5 20930	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
73	72	sseld 3933	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ (𝑁‘{𝑋}) → 𝐸 ∈ (𝑁‘{𝑋, 𝑌})))
74	73	con3dimp 408	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝐸 ∈ (𝑁‘{𝑋}))
75	74	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝐸 ∈ (𝑁‘{𝑋}))
76	3, 23, 4, 68, 57, 71, 75	lspsnnecom 21057	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝐸}))
77	3, 49, 10, 53, 62, 65, 66, 76	lssvancl1 20879	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸}))
78		eleq1 2819	. . . . . . . 8 ⊢ (𝑧 = (𝐸 + 𝑋) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌})))
79	78	notbid 318	. . . . . . 7 ⊢ (𝑧 = (𝐸 + 𝑋) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌})))
80		eleq1 2819	. . . . . . . 8 ⊢ (𝑧 = (𝐸 + 𝑋) → (𝑧 ∈ (𝑁‘{𝐸}) ↔ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸})))
81	80	notbid 318	. . . . . . 7 ⊢ (𝑧 = (𝐸 + 𝑋) → (¬ 𝑧 ∈ (𝑁‘{𝐸}) ↔ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸})))
82	79, 81	anbi12d 632	. . . . . 6 ⊢ (𝑧 = (𝐸 + 𝑋) → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})) ↔ (¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸}))))
83	82	rspcev 3577	. . . . 5 ⊢ (((𝐸 + 𝑋) ∈ 𝑉 ∧ (¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸}))) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
84	52, 59, 77, 83	syl12anc 836	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
85	48, 84	pm2.61dane 3015	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
86	20, 85	pm2.61dan 812	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
87		hdmap11.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
88		hdmap11.a	. . . 4 ⊢ ✚ = (+_g‘𝐶)
89		hdmap11.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
90	5	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
91	6	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑋 ∈ 𝑉)
92	7	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑌 ∈ 𝑉)
93		hdmap11.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
94		hdmap11.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
95		hdmap11.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
96		hdmap11.j	. . . 4 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊)
97		hdmap11.i	. . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊)
98		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑧 ∈ 𝑉)
99		simp3l 1202	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
100	11	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑈 ∈ LMod)
101	26	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝐸 ∈ 𝑉)
102		simp3r 1203	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → ¬ 𝑧 ∈ (𝑁‘{𝐸}))
103	3, 4, 100, 98, 101, 102	lspsnne2 21056	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝐸}))
104	1, 2, 3, 49, 87, 88, 89, 90, 91, 92, 24, 23, 4, 93, 94, 95, 96, 97, 98, 99, 103	hdmap11lem1 41886	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
105	104	rexlimdv3a 3137	. 2 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})) → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))))
106	86, 105	mpd 15	1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 ∖ cdif 3899 ⊆ wss 3902 {csn 4576 {cpr 4578 ⟨cop 4582 I cid 5510 ↾ cres 5618 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 LModclmod 20794 LSubSpclss 20865 LSpanclspn 20905 LVecclvec 21037 HLchlt 39395 LHypclh 40029 LTrncltrn 40146 DVecHcdvh 41123 LCDualclcd 41631 mapdcmpd 41669 HVMapchvm 41801 HDMap1chdma1 41836 HDMapchdma 41837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 38998

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-ot 4585 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19230 df-oppg 19259 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20796 df-lss 20866 df-lsp 20906 df-lvec 21038 df-lsatoms 39021 df-lshyp 39022 df-lcv 39064 df-lfl 39103 df-lkr 39131 df-ldual 39169 df-oposet 39221 df-ol 39223 df-oml 39224 df-covers 39311 df-ats 39312 df-atl 39343 df-cvlat 39367 df-hlat 39396 df-llines 39543 df-lplanes 39544 df-lvols 39545 df-lines 39546 df-psubsp 39548 df-pmap 39549 df-padd 39841 df-lhyp 40033 df-laut 40034 df-ldil 40149 df-ltrn 40150 df-trl 40204 df-tgrp 40788 df-tendo 40800 df-edring 40802 df-dveca 41048 df-disoa 41074 df-dvech 41124 df-dib 41184 df-dic 41218 df-dih 41274 df-doch 41393 df-djh 41440 df-lcdual 41632 df-mapd 41670 df-hvmap 41802 df-hdmap1 41838 df-hdmap 41839

This theorem is referenced by: hdmapadd 41888

W3C validator