Theorem hdmaprnlem3eN 42521

Description: Lemma for hdmaprnN 42527. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hdmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hdmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)
hdmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmaprnlem1.se	⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
hdmaprnlem1.ve	⊢ (𝜑 → 𝑣 ∈ 𝑉)
hdmaprnlem1.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
hdmaprnlem1.ue	⊢ (𝜑 → 𝑢 ∈ 𝑉)
hdmaprnlem1.un	⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
hdmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hdmaprnlem1.q	⊢ 𝑄 = (0g‘𝐶)
hdmaprnlem1.o	⊢ 0 = (0g‘𝑈)
hdmaprnlem1.a	⊢ ✚ = (+g‘𝐶)
hdmaprnlem3e.p	⊢ + = (+g‘𝑈)

Assertion

Ref	Expression
hdmaprnlem3eN	⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))

Distinct variable groups:   𝑡, ✚   𝑡,𝐿   𝑡,𝑀   𝑡,𝑁   𝑡, 0   𝑡, +   𝑡,𝑆   𝑡,𝑈   𝑡,𝑉   𝜑,𝑡   𝑡,𝑠,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑣,𝑢,𝑠)   𝐶(𝑣,𝑢,𝑡,𝑠)   𝐷(𝑣,𝑢,𝑡,𝑠)   + (𝑣,𝑢,𝑠)   ✚ (𝑣,𝑢,𝑠)   𝑄(𝑣,𝑢,𝑡,𝑠)   𝑆(𝑣,𝑢,𝑠)   𝑈(𝑣,𝑢,𝑠)   𝐻(𝑣,𝑢,𝑡,𝑠)   𝐾(𝑣,𝑢,𝑡,𝑠)   𝐿(𝑣,𝑢,𝑠)   𝑀(𝑣,𝑢,𝑠)   𝑁(𝑣,𝑢,𝑠)   𝑉(𝑣,𝑢,𝑠)   𝑊(𝑣,𝑢,𝑡,𝑠)   0 (𝑣,𝑢,𝑠)

Proof of Theorem hdmaprnlem3eN

Step	Hyp	Ref	Expression
1		hdmaprnlem1.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
2		hdmaprnlem3e.p	. . 3 ⊢ + = (+g‘𝑈)
3		hdmaprnlem1.o	. . 3 ⊢ 0 = (0g‘𝑈)
4		hdmaprnlem1.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
5		eqid 2769	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
6		hdmaprnlem1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		hdmaprnlem1.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		hdmaprnlem1.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	6, 7, 8	dvhlvec 41772	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
10		hdmaprnlem1.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
11		hdmaprnlem1.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		eqid 2769	. . . 4 ⊢ (LSAtoms‘𝐶) = (LSAtoms‘𝐶)
13		hdmaprnlem1.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
14		hdmaprnlem1.l	. . . . 5 ⊢ 𝐿 = (LSpan‘𝐶)
15		hdmaprnlem1.q	. . . . 5 ⊢ 𝑄 = (0g‘𝐶)
16	6, 11, 8	lcdlmod 42255	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
17		hdmaprnlem1.s	. . . . . . . 8 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
18		hdmaprnlem1.ue	. . . . . . . 8 ⊢ (𝜑 → 𝑢 ∈ 𝑉)
19	6, 7, 1, 11, 13, 17, 8, 18	hdmapcl 42493	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
20		hdmaprnlem1.se	. . . . . . . 8 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
21	20	eldifad 3925	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
22		hdmaprnlem1.a	. . . . . . . 8 ⊢ ✚ = (+g‘𝐶)
23	13, 22	lmodvacl 20973	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
24	16, 19, 21, 23	syl3anc 1396	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
25		hdmaprnlem1.ve	. . . . . . . 8 ⊢ (𝜑 → 𝑣 ∈ 𝑉)
26		hdmaprnlem1.e	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
27		hdmaprnlem1.un	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
28	6, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27	hdmaprnlem1N 42512	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
29	13, 22, 15, 14, 16, 19, 21, 28	lmodindp1 21112	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ≠ 𝑄)
30		eldifsn 4758	. . . . . 6 ⊢ (((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐷 ∖ {𝑄}) ↔ (((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷 ∧ ((𝑆‘𝑢) ✚ 𝑠) ≠ 𝑄))
31	24, 29, 30	sylanbrc 594	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐷 ∖ {𝑄}))
32	13, 14, 15, 12, 16, 31	lsatlspsn 39656	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSAtoms‘𝐶))
33	6, 10, 7, 5, 11, 12, 8, 32	mapdcnvatN 42329	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ∈ (LSAtoms‘𝑈))
34	6, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27, 13, 15, 3, 22	hdmaprnlem3uN 42514	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
35	34	necomd 3019	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ≠ (𝑁‘{𝑢}))
36	6, 7, 1, 4, 11, 14, 10, 17, 8, 20, 25, 26, 18, 27, 13, 15, 3, 22	hdmaprnlem3N 42513	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
37	36	necomd 3019	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ≠ (𝑁‘{𝑣}))
38		eqid 2769	. . . . . . 7 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
39		eqid 2769	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
40	6, 7, 8	dvhlmod 41773	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
41	1, 39, 4	lspsncl 21075	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉) → (𝑁‘{𝑢}) ∈ (LSubSp‘𝑈))
42	40, 18, 41	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑢}) ∈ (LSubSp‘𝑈))
43	6, 10, 7, 39, 11, 38, 8, 42	mapdcl2 42319	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) ∈ (LSubSp‘𝐶))
44	1, 39, 4	lspsncl 21075	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
45	40, 25, 44	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
46	6, 10, 7, 39, 11, 38, 8, 45	mapdcl2 42319	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ∈ (LSubSp‘𝐶))
47		eqid 2769	. . . . . . . . 9 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
48	38, 47	lsmcl 21181	. . . . . . . 8 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑢})) ∈ (LSubSp‘𝐶) ∧ (𝑀‘(𝑁‘{𝑣})) ∈ (LSubSp‘𝐶)) → ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))) ∈ (LSubSp‘𝐶))
49	16, 43, 46, 48	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))) ∈ (LSubSp‘𝐶))
50	38	lsssssubg 21056	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
51	16, 50	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
52	51, 43	sseldd 3946	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) ∈ (SubGrp‘𝐶))
53	51, 46	sseldd 3946	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ∈ (SubGrp‘𝐶))
54	13, 14	lspsnid 21091	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷) → (𝑆‘𝑢) ∈ (𝐿‘{(𝑆‘𝑢)}))
55	16, 19, 54	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝐿‘{(𝑆‘𝑢)}))
56	6, 7, 1, 4, 11, 14, 10, 17, 8, 18	hdmap10 42503	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐿‘{(𝑆‘𝑢)}))
57	55, 56	eleqtrrd 2872	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝑀‘(𝑁‘{𝑢})))
58		eqimss2 4004	. . . . . . . . . 10 ⊢ ((𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}) → (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣})))
59	26, 58	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣})))
60	13, 38, 14, 16, 46, 21	ellspsn5b 21093	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝑀‘(𝑁‘{𝑣})) ↔ (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣}))))
61	59, 60	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑣})))
62	22, 47	lsmelvali 19719	. . . . . . . 8 ⊢ ((((𝑀‘(𝑁‘{𝑢})) ∈ (SubGrp‘𝐶) ∧ (𝑀‘(𝑁‘{𝑣})) ∈ (SubGrp‘𝐶)) ∧ ((𝑆‘𝑢) ∈ (𝑀‘(𝑁‘{𝑢})) ∧ 𝑠 ∈ (𝑀‘(𝑁‘{𝑣})))) → ((𝑆‘𝑢) ✚ 𝑠) ∈ ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
63	52, 53, 57, 61, 62	syl22anc 851	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
64	38, 14, 16, 49, 63	ellspsn5 21094	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ⊆ ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
65		eqid 2769	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
66	6, 10, 7, 39, 65, 11, 47, 8, 42, 45	mapdlsm 42327	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))) = ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
67	64, 66	sseqtrrd 3982	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ⊆ (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))))
68	13, 38, 14	lspsncl 21075	. . . . . . . 8 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
69	16, 24, 68	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
70	6, 10, 11, 38, 8	mapdrn2 42314	. . . . . . 7 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
71	69, 70	eleqtrrd 2872	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
72	39, 65	lsmcl 21181	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑢}) ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈)) → ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})) ∈ (LSubSp‘𝑈))
73	40, 42, 45, 72	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})) ∈ (LSubSp‘𝑈))
74	6, 10, 7, 39, 8, 73	mapdcl 42316	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))) ∈ ran 𝑀)
75	6, 10, 8, 71, 74	mapdcnvordN 42321	. . . . 5 ⊢ (𝜑 → ((◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ⊆ (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))) ↔ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ⊆ (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))))
76	67, 75	mpbird 260	. . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ⊆ (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))))
77	1, 4, 65, 40, 18, 25	lsmpr 21187	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢, 𝑣}) = ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))
78	6, 10, 7, 39, 8, 73	mapdcnvid1N 42317	. . . . 5 ⊢ (𝜑 → (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))) = ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))
79	77, 78	eqtr4d 2807	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑢, 𝑣}) = (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))))
80	76, 79	sseqtrrd 3982	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ⊆ (𝑁‘{𝑢, 𝑣}))
81	1, 2, 3, 4, 5, 9, 33, 18, 25, 35, 37, 80	lsatfixedN 39672	. 2 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)}))
82		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)}))
83	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
84	40	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑈 ∈ LMod)
85	18	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑢 ∈ 𝑉)
86	20	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑠 ∈ (𝐷 ∖ {𝑄}))
87	25	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑣 ∈ 𝑉)
88	26	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
89	27	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
90		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))
91	6, 7, 1, 4, 11, 14, 10, 17, 83, 86, 87, 88, 85, 89, 13, 15, 3, 22, 90	hdmaprnlem4tN 42515	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑡 ∈ 𝑉)
92	1, 2	lmodvacl 20973	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝑢 + 𝑡) ∈ 𝑉)
93	84, 85, 91, 92	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑢 + 𝑡) ∈ 𝑉)
94	1, 39, 4	lspsncl 21075	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝑢 + 𝑡) ∈ 𝑉) → (𝑁‘{(𝑢 + 𝑡)}) ∈ (LSubSp‘𝑈))
95	84, 93, 94	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑁‘{(𝑢 + 𝑡)}) ∈ (LSubSp‘𝑈))
96	6, 10, 7, 39, 83, 95	mapdcnvid1N 42317	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (◡𝑀‘(𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) = (𝑁‘{(𝑢 + 𝑡)}))
97	82, 96	eqtr4d 2807	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (◡𝑀‘(𝑀‘(𝑁‘{(𝑢 + 𝑡)}))))
98	71	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
99	6, 10, 7, 39, 83, 95	mapdcl 42316	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑀‘(𝑁‘{(𝑢 + 𝑡)})) ∈ ran 𝑀)
100	6, 10, 83, 98, 99	mapdcnv11N 42322	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → ((◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (◡𝑀‘(𝑀‘(𝑁‘{(𝑢 + 𝑡)}))) ↔ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))))
101	97, 100	mpbid 235	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
102	101	ex 417	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) → ((◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)}) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))))
103	102	reximdva 3184	. 2 ⊢ (𝜑 → (∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)}) → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)}))))
104	81, 103	mpd 16	1 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  ^◡ccnv 5661  ran crn 5663  ‘cfv 6537  (class class class)co 7411  Basecbs 17268  +_gcplusg 17309  0_gc0g 17491  SubGrpcsubg 19185  LSSumclsm 19703  LModclmod 20958  LSubSpclss 21029  LSpanclspn 21069  LSAtomsclsa 39637  HLchlt 40013  LHypclh 40647  DVecHcdvh 41741  LCDualclcd 42249  mapdcmpd 42287  HDMapchdma 42455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-riotaBAD 39616

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-undef 8268  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-n0 12504  df-z 12591  df-uz 12862  df-fz 13535  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-sca 17325  df-vsca 17326  df-0g 17493  df-mre 17637  df-mrc 17638  df-acs 17640  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-p1 18479  df-lat 18487  df-clat 18554  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-grp 19002  df-minusg 19003  df-sbg 19004  df-subg 19188  df-cntz 19386  df-oppg 19415  df-lsm 19705  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-oppr 20418  df-dvdsr 20438  df-unit 20439  df-invr 20469  df-dvr 20482  df-nzr 20595  df-rlreg 20778  df-domn 20779  df-drng 20814  df-lmod 20960  df-lss 21030  df-lsp 21070  df-lvec 21201  df-lsatoms 39639  df-lshyp 39640  df-lcv 39682  df-lfl 39721  df-lkr 39749  df-ldual 39787  df-oposet 39839  df-ol 39841  df-oml 39842  df-covers 39929  df-ats 39930  df-atl 39961  df-cvlat 39985  df-hlat 40014  df-llines 40161  df-lplanes 40162  df-lvols 40163  df-lines 40164  df-psubsp 40166  df-pmap 40167  df-padd 40459  df-lhyp 40651  df-laut 40652  df-ldil 40767  df-ltrn 40768  df-trl 40822  df-tgrp 41406  df-tendo 41418  df-edring 41420  df-dveca 41666  df-disoa 41692  df-dvech 41742  df-dib 41802  df-dic 41836  df-dih 41892  df-doch 42011  df-djh 42058  df-lcdual 42250  df-mapd 42288  df-hvmap 42420  df-hdmap1 42456  df-hdmap 42457

This theorem is referenced by:  hdmaprnlem10N  42522