hdmap11lem2 - Mathbox for Norm Megill

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Theorem hdmap11lem2 42473

Description: Lemma for hdmapadd 42474. (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 42077	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
10		eqid 2765	. . . . . . . 8 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
11	1, 2, 5	dvhlmod 41741	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	11	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod)
13	3, 10, 4, 11, 6, 7	lspprcl 21065	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
14	13	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
15		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐸 ∈ (𝑁‘{𝑋, 𝑌}))
16	10, 4, 12, 14, 15	ellspsn5 21083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝑋, 𝑌}))
17	16	ssneld 3941	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝐸})))
18	17	ancld 559	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
19	18	reximdv 3180	. . . 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 16	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
21		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
22		eqid 2765	. . . . . . . . . 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 41743	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))
26	25	eldifad 3919	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
27	1, 2, 3, 4, 5, 26, 7	dvh3dim 42077	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}))
28	27	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}))
29		preq1 4695	. . . . . . . . . . . . 13 ⊢ (𝑋 = 0 → {𝑋, 𝑌} = { 0 , 𝑌})
30		prcom 4694	. . . . . . . . . . . . 13 ⊢ { 0 , 𝑌} = {𝑌, 0 }
31	29, 30	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ (𝑋 = 0 → {𝑋, 𝑌} = {𝑌, 0 })
32	31	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑋 = 0 → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, 0 }))
33	3, 23, 4, 11, 7	lsppr0 21179	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌, 0 }) = (𝑁‘{𝑌}))
34	32, 33	sylan9eqr 2822	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌}))
35	3, 10, 4, 11, 26, 7	lspprcl 21065	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑌}) ∈ (LSubSp‘𝑈))
36	3, 4, 11, 26, 7	lspprid2 21085	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝐸, 𝑌}))
37	10, 4, 11, 35, 36	ellspsn5 21083	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
38	37	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
39	34, 38	eqsstrd 3973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝐸, 𝑌}))
40	39	ssneld 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
41	3, 4, 11, 26, 7	lspprid1 21084	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑌}))
42	10, 4, 11, 35, 41	ellspsn5 21083	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑌}))
43	42	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑌}))
44	43	ssneld 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ¬ 𝑧 ∈ (𝑁‘{𝐸})))
45	40, 44	jcad 521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
46	45	reximdv 3180	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝐸, 𝑌}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))))
47	28, 46	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 0 ) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
48	47	adantlr 727	. . . 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 20962	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) → (𝐸 + 𝑋) ∈ 𝑉)
51	11, 26, 6, 50	syl3anc 1394	. . . . . 6 ⊢ (𝜑 → (𝐸 + 𝑋) ∈ 𝑉)
52	51	ad2antrr 738	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝐸 + 𝑋) ∈ 𝑉)
53	11	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑈 ∈ LMod)
54	13	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
55	3, 4, 11, 6, 7	lspprid1 21084	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑌}))
56	55	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑁‘{𝑋, 𝑌}))
57	26	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝐸 ∈ 𝑉)
58		simplr 780	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌}))
59	3, 49, 10, 53, 54, 56, 57, 58	lssvancl2 21033	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌}))
60	3, 10, 4	lspsncl 21064	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
61	11, 26, 60	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
62	61	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
63	3, 4	lspsnid 21080	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → 𝐸 ∈ (𝑁‘{𝐸}))
64	11, 26, 63	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸}))
65	64	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝐸 ∈ (𝑁‘{𝐸}))
66	6	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝑉)
67	1, 2, 5	dvhlvec 41740	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
68	67	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑈 ∈ LVec)
69		simpr 489	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 )
70		eldifsn 4749	. . . . . . . 8 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
71	66, 69, 70	sylanbrc 594	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
72	10, 4, 11, 13, 55	ellspsn5 21083	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑌}))
73	72	sseld 3938	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ (𝑁‘{𝑋}) → 𝐸 ∈ (𝑁‘{𝑋, 𝑌})))
74	73	con3dimp 413	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝐸 ∈ (𝑁‘{𝑋}))
75	74	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝐸 ∈ (𝑁‘{𝑋}))
76	3, 23, 4, 68, 57, 71, 75	lspsnnecom 21209	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝐸}))
77	3, 49, 10, 53, 62, 65, 66, 76	lssvancl1 21032	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸}))
78		eleq1 2853	. . . . . . . 8 ⊢ (𝑧 = (𝐸 + 𝑋) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌})))
79	78	notbid 321	. . . . . . 7 ⊢ (𝑧 = (𝐸 + 𝑋) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌})))
80		eleq1 2853	. . . . . . . 8 ⊢ (𝑧 = (𝐸 + 𝑋) → (𝑧 ∈ (𝑁‘{𝐸}) ↔ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸})))
81	80	notbid 321	. . . . . . 7 ⊢ (𝑧 = (𝐸 + 𝑋) → (¬ 𝑧 ∈ (𝑁‘{𝐸}) ↔ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸})))
82	79, 81	anbi12d 643	. . . . . 6 ⊢ (𝑧 = (𝐸 + 𝑋) → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})) ↔ (¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝐸 + 𝑋) ∈ (𝑁‘{𝐸}))))
83	82	rspcev 3584	. . . . 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 849	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) ∧ 𝑋 ≠ 0 ) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
85	48, 84	pm2.61dane 3047	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐸 ∈ (𝑁‘{𝑋, 𝑌})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})))
86	20, 85	pm2.61dan 824	. 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 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
91	6	3ad2ant1 1149	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑋 ∈ 𝑉)
92	7	3ad2ant1 1149	. . . 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 1153	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑧 ∈ 𝑉)
99		simp3l 1218	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
100	11	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝑈 ∈ LMod)
101	26	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → 𝐸 ∈ 𝑉)
102		simp3r 1219	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → ¬ 𝑧 ∈ (𝑁‘{𝐸}))
103	3, 4, 100, 98, 101, 102	lspsnne2 21208	. . . 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 42472	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉 ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸}))) → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
105	104	rexlimdv3a 3170	. 2 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝐸})) → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))))
106	86, 105	mpd 16	1 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∃wrex 3089 ∖ cdif 3904 ⊆ wss 3907 {csn 4585 {cpr 4587 ⟨cop 4591 I cid 5545 ↾ cres 5653 ‘cfv 6525 (class class class)co 7400 Basecbs 17257 +_gcplusg 17298 0_gc0g 17480 LModclmod 20947 LSubSpclss 21018 LSpanclspn 21058 LVecclvec 21189 HLchlt 39981 LHypclh 40615 LTrncltrn 40732 DVecHcdvh 41709 LCDualclcd 42217 mapdcmpd 42255 HVMapchvm 42387 HDMap1chdma1 42422 HDMapchdma 42423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-riotaBAD 39584

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-n0 12493 df-z 12580 df-uz 12851 df-fz 13524 df-struct 17195 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-ress 17279 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17482 df-mre 17626 df-mrc 17627 df-acs 17629 df-proset 18338 df-poset 18357 df-plt 18372 df-lub 18388 df-glb 18389 df-join 18390 df-meet 18391 df-p0 18467 df-p1 18468 df-lat 18476 df-clat 18543 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-submnd 18830 df-grp 18991 df-minusg 18992 df-sbg 18993 df-subg 19177 df-cntz 19375 df-oppg 19404 df-lsm 19694 df-cmn 19840 df-abl 19841 df-mgp 20205 df-rng 20219 df-ur 20252 df-ring 20305 df-oppr 20407 df-dvdsr 20427 df-unit 20428 df-invr 20458 df-dvr 20471 df-nzr 20584 df-rlreg 20767 df-domn 20768 df-drng 20803 df-lmod 20949 df-lss 21019 df-lsp 21059 df-lvec 21190 df-lsatoms 39607 df-lshyp 39608 df-lcv 39650 df-lfl 39689 df-lkr 39717 df-ldual 39755 df-oposet 39807 df-ol 39809 df-oml 39810 df-covers 39897 df-ats 39898 df-atl 39929 df-cvlat 39953 df-hlat 39982 df-llines 40129 df-lplanes 40130 df-lvols 40131 df-lines 40132 df-psubsp 40134 df-pmap 40135 df-padd 40427 df-lhyp 40619 df-laut 40620 df-ldil 40735 df-ltrn 40736 df-trl 40790 df-tgrp 41374 df-tendo 41386 df-edring 41388 df-dveca 41634 df-disoa 41660 df-dvech 41710 df-dib 41770 df-dic 41804 df-dih 41860 df-doch 41979 df-djh 42026 df-lcdual 42218 df-mapd 42256 df-hvmap 42388 df-hdmap1 42424 df-hdmap 42425

This theorem is referenced by: hdmapadd 42474

W3C validator