Theorem hdmaprnlem3eN 42479

Description: Lemma for hdmaprnN 42485. (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 2762	. . 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 41730	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
10		hdmaprnlem1.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
11		hdmaprnlem1.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		eqid 2762	. . . 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 42213	. . . . 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 42451	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑢) ∈ 𝐷)
20		hdmaprnlem1.se	. . . . . . . 8 ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))
21	20	eldifad 3916	. . . . . . 7 ⊢ (𝜑 → 𝑠 ∈ 𝐷)
22		hdmaprnlem1.a	. . . . . . . 8 ⊢ ✚ = (+g‘𝐶)
23	13, 22	lmodvacl 20939	. . . . . . 7 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷) → ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷)
24	16, 19, 21, 23	syl3anc 1390	. . . . . 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 42470	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
29	13, 22, 15, 14, 16, 19, 21, 28	lmodindp1 21078	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ≠ 𝑄)
30		eldifsn 4746	. . . . . 6 ⊢ (((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐷 ∖ {𝑄}) ↔ (((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷 ∧ ((𝑆‘𝑢) ✚ 𝑠) ≠ 𝑄))
31	24, 29, 30	sylanbrc 592	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ (𝐷 ∖ {𝑄}))
32	13, 14, 15, 12, 16, 31	lsatlspsn 39614	. . . 4 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSAtoms‘𝐶))
33	6, 10, 7, 5, 11, 12, 8, 32	mapdcnvatN 42287	. . 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 42472	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
35	34	necomd 3012	. . 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 42471	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
37	36	necomd 3012	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ≠ (𝑁‘{𝑣}))
38		eqid 2762	. . . . . . 7 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
39		eqid 2762	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
40	6, 7, 8	dvhlmod 41731	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
41	1, 39, 4	lspsncl 21041	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉) → (𝑁‘{𝑢}) ∈ (LSubSp‘𝑈))
42	40, 18, 41	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑢}) ∈ (LSubSp‘𝑈))
43	6, 10, 7, 39, 11, 38, 8, 42	mapdcl2 42277	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) ∈ (LSubSp‘𝐶))
44	1, 39, 4	lspsncl 21041	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉) → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
45	40, 25, 44	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑣}) ∈ (LSubSp‘𝑈))
46	6, 10, 7, 39, 11, 38, 8, 45	mapdcl2 42277	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ∈ (LSubSp‘𝐶))
47		eqid 2762	. . . . . . . . 9 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
48	38, 47	lsmcl 21147	. . . . . . . 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 1390	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))) ∈ (LSubSp‘𝐶))
50	38	lsssssubg 21022	. . . . . . . . . 10 ⊢ (𝐶 ∈ LMod → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
51	16, 50	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (LSubSp‘𝐶) ⊆ (SubGrp‘𝐶))
52	51, 43	sseldd 3937	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) ∈ (SubGrp‘𝐶))
53	51, 46	sseldd 3937	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) ∈ (SubGrp‘𝐶))
54	13, 14	lspsnid 21057	. . . . . . . . . 10 ⊢ ((𝐶 ∈ LMod ∧ (𝑆‘𝑢) ∈ 𝐷) → (𝑆‘𝑢) ∈ (𝐿‘{(𝑆‘𝑢)}))
55	16, 19, 54	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝐿‘{(𝑆‘𝑢)}))
56	6, 7, 1, 4, 11, 14, 10, 17, 8, 18	hdmap10 42461	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑢})) = (𝐿‘{(𝑆‘𝑢)}))
57	55, 56	eleqtrrd 2865	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑢) ∈ (𝑀‘(𝑁‘{𝑢})))
58		eqimss2 3995	. . . . . . . . . 10 ⊢ ((𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}) → (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣})))
59	26, 58	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣})))
60	13, 38, 14, 16, 46, 21	ellspsn5b 21059	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ (𝑀‘(𝑁‘{𝑣})) ↔ (𝐿‘{𝑠}) ⊆ (𝑀‘(𝑁‘{𝑣}))))
61	59, 60	mpbird 259	. . . . . . . 8 ⊢ (𝜑 → 𝑠 ∈ (𝑀‘(𝑁‘{𝑣})))
62	22, 47	lsmelvali 19690	. . . . . . . 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 849	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑢) ✚ 𝑠) ∈ ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
64	38, 14, 16, 49, 63	ellspsn5 21060	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ⊆ ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
65		eqid 2762	. . . . . . 7 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
66	6, 10, 7, 39, 65, 11, 47, 8, 42, 45	mapdlsm 42285	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))) = ((𝑀‘(𝑁‘{𝑢}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑣}))))
67	64, 66	sseqtrrd 3973	. . . . 5 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ⊆ (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))))
68	13, 38, 14	lspsncl 21041	. . . . . . . 8 ⊢ ((𝐶 ∈ LMod ∧ ((𝑆‘𝑢) ✚ 𝑠) ∈ 𝐷) → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
69	16, 24, 68	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ (LSubSp‘𝐶))
70	6, 10, 11, 38, 8	mapdrn2 42272	. . . . . . 7 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐶))
71	69, 70	eleqtrrd 2865	. . . . . 6 ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) ∈ ran 𝑀)
72	39, 65	lsmcl 21147	. . . . . . . 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 1390	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})) ∈ (LSubSp‘𝑈))
74	6, 10, 7, 39, 8, 73	mapdcl 42274	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣}))) ∈ ran 𝑀)
75	6, 10, 8, 71, 74	mapdcnvordN 42279	. . . . 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 259	. . . 4 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ⊆ (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))))
77	1, 4, 65, 40, 18, 25	lsmpr 21153	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑢, 𝑣}) = ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))
78	6, 10, 7, 39, 8, 73	mapdcnvid1N 42275	. . . . 5 ⊢ (𝜑 → (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))) = ((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))
79	77, 78	eqtr4d 2800	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑢, 𝑣}) = (◡𝑀‘(𝑀‘((𝑁‘{𝑢})(LSSum‘𝑈)(𝑁‘{𝑣})))))
80	76, 79	sseqtrrd 3973	. . 3 ⊢ (𝜑 → (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) ⊆ (𝑁‘{𝑢, 𝑣}))
81	1, 2, 3, 4, 5, 9, 33, 18, 25, 35, 37, 80	lsatfixedN 39630	. 2 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)}))
82		simpr 488	. . . . . 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 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
84	40	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑈 ∈ LMod)
85	18	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑢 ∈ 𝑉)
86	20	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑠 ∈ (𝐷 ∖ {𝑄}))
87	25	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑣 ∈ 𝑉)
88	26	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))
89	27	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → ¬ 𝑢 ∈ (𝑁‘{𝑣}))
90		simplr 778	. . . . . . . . . 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 42473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → 𝑡 ∈ 𝑉)
92	1, 2	lmodvacl 20939	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑢 ∈ 𝑉 ∧ 𝑡 ∈ 𝑉) → (𝑢 + 𝑡) ∈ 𝑉)
93	84, 85, 91, 92	syl3anc 1390	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })) ∧ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) = (𝑁‘{(𝑢 + 𝑡)})) → (𝑢 + 𝑡) ∈ 𝑉)
94	1, 39, 4	lspsncl 21041	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ (𝑢 + 𝑡) ∈ 𝑉) → (𝑁‘{(𝑢 + 𝑡)}) ∈ (LSubSp‘𝑈))
95	84, 93, 94	syl2anc 593	. . . . . . 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 42275	. . . . . 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 2800	. . . . 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 736	. . . . . 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 42274	. . . . . 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 42280	. . . . 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 234	. . . 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 416	. . 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 3175	. 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 15	1 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086   ∖ cdif 3901   ⊆ wss 3904  {csn 4582  {cpr 4584  ^◡ccnv 5646  ran crn 5648  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  0_gc0g 17468  SubGrpcsubg 19162  LSSumclsm 19674  LModclmod 20924  LSubSpclss 20995  LSpanclspn 21035  LSAtomsclsa 39595  HLchlt 39971  LHypclh 40605  DVecHcdvh 41699  LCDualclcd 42207  mapdcmpd 42245  HDMapchdma 42413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-riotaBAD 39574

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-tpos 8206  df-undef 8253  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-n0 12482  df-z 12569  df-uz 12840  df-fz 13513  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-0g 17470  df-mre 17614  df-mrc 17615  df-acs 17617  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p0 18455  df-p1 18456  df-lat 18464  df-clat 18531  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-subg 19165  df-cntz 19357  df-oppg 19386  df-lsm 19676  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20228  df-ring 20281  df-oppr 20382  df-dvdsr 20402  df-unit 20403  df-invr 20433  df-dvr 20446  df-nzr 20559  df-rlreg 20740  df-domn 20741  df-drng 20777  df-lmod 20926  df-lss 20996  df-lsp 21036  df-lvec 21167  df-lsatoms 39597  df-lshyp 39598  df-lcv 39640  df-lfl 39679  df-lkr 39707  df-ldual 39745  df-oposet 39797  df-ol 39799  df-oml 39800  df-covers 39887  df-ats 39888  df-atl 39919  df-cvlat 39943  df-hlat 39972  df-llines 40119  df-lplanes 40120  df-lvols 40121  df-lines 40122  df-psubsp 40124  df-pmap 40125  df-padd 40417  df-lhyp 40609  df-laut 40610  df-ldil 40725  df-ltrn 40726  df-trl 40780  df-tgrp 41364  df-tendo 41376  df-edring 41378  df-dveca 41624  df-disoa 41650  df-dvech 41700  df-dib 41760  df-dic 41794  df-dih 41850  df-doch 41969  df-djh 42016  df-lcdual 42208  df-mapd 42246  df-hvmap 42378  df-hdmap1 42414  df-hdmap 42415

This theorem is referenced by:  hdmaprnlem10N  42480