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Theorem pjthlem2 25406

Description: Lemma for pjth 25407. (Contributed by NM, 10-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.)

**Hypotheses**
Ref	Expression
pjthlem.v	⊢ 𝑉 = (Base‘𝑊)
pjthlem.n	⊢ 𝑁 = (norm‘𝑊)
pjthlem.p	⊢ + = (+_g‘𝑊)
pjthlem.m	⊢ − = (-_g‘𝑊)
pjthlem.h	⊢ , = (·_𝑖‘𝑊)
pjthlem.l	⊢ 𝐿 = (LSubSp‘𝑊)
pjthlem.1	⊢ (𝜑 → 𝑊 ∈ ℂHil)
pjthlem.2	⊢ (𝜑 → 𝑈 ∈ 𝐿)
pjthlem.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pjthlem.j	⊢ 𝐽 = (TopOpen‘𝑊)
pjthlem.s	⊢ ⊕ = (LSSum‘𝑊)
pjthlem.o	⊢ 𝑂 = (ocv‘𝑊)
pjthlem.3	⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽))

**Assertion**
Ref	Expression
pjthlem2	⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))

Proof of Theorem pjthlem2 Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjthlem.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		pjthlem.m	. . . 4 ⊢ − = (-_g‘𝑊)
3		pjthlem.n	. . . 4 ⊢ 𝑁 = (norm‘𝑊)
4		pjthlem.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂHil)
5		hlcph 25332	. . . . 5 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil)
6	4, 5	syl 17	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil)
7		pjthlem.2	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
8		pjthlem.l	. . . . 5 ⊢ 𝐿 = (LSubSp‘𝑊)
9	7, 8	eleqtrdi 2847	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊))
10		pjthlem.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽))
11		hlcms 25334	. . . . . . 7 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
12	4, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CMetSp)
13	1, 8	lssss 20899	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ 𝑉)
14	7, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
15		eqid 2737	. . . . . . 7 ⊢ (𝑊 ↾_s 𝑈) = (𝑊 ↾_s 𝑈)
16		pjthlem.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
17	15, 1, 16	cmsss 25319	. . . . . 6 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑈 ⊆ 𝑉) → ((𝑊 ↾_s 𝑈) ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽)))
18	12, 14, 17	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((𝑊 ↾_s 𝑈) ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽)))
19	10, 18	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑊 ↾_s 𝑈) ∈ CMetSp)
20		pjthlem.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
21	1, 2, 3, 6, 9, 19, 20	minvec 25404	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
22		reurex 3356	. . 3 ⊢ (∃!𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) → ∃𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
23	21, 22	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
24	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑊 ∈ ℂPreHil)
25		cphlmod 25142	. . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
26	24, 25	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑊 ∈ LMod)
27		lmodabl 20872	. . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
28	26, 27	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑊 ∈ Abel)
29	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑈 ∈ 𝐿)
30	29, 13	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑈 ⊆ 𝑉)
31		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑥 ∈ 𝑈)
32	30, 31	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑥 ∈ 𝑉)
33	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝐴 ∈ 𝑉)
34		pjthlem.p	. . . . 5 ⊢ + = (+_g‘𝑊)
35	1, 34, 2	ablpncan3 19757	. . . 4 ⊢ ((𝑊 ∈ Abel ∧ (𝑥 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → (𝑥 + (𝐴 − 𝑥)) = 𝐴)
36	28, 32, 33, 35	syl12anc 837	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑥 + (𝐴 − 𝑥)) = 𝐴)
37	8	lsssssubg 20921	. . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊))
38	26, 37	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝐿 ⊆ (SubGrp‘𝑊))
39	38, 29	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑈 ∈ (SubGrp‘𝑊))
40		cphphl 25139	. . . . . . 7 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
41	24, 40	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑊 ∈ PreHil)
42		pjthlem.o	. . . . . . 7 ⊢ 𝑂 = (ocv‘𝑊)
43	1, 42, 8	ocvlss 21639	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ⊆ 𝑉) → (𝑂‘𝑈) ∈ 𝐿)
44	41, 30, 43	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ 𝐿)
45	38, 44	sseldd 3936	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ (SubGrp‘𝑊))
46	1, 2	lmodvsubcl 20870	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝐴 − 𝑥) ∈ 𝑉)
47	26, 33, 32, 46	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝐴 − 𝑥) ∈ 𝑉)
48		pjthlem.h	. . . . . . . 8 ⊢ , = (·_𝑖‘𝑊)
49	4	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ ℂHil)
50	29	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝐿)
51	47	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → (𝐴 − 𝑥) ∈ 𝑉)
52		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
53		oveq2 7376	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝐴 − 𝑦) = (𝐴 − (𝑤 + 𝑥)))
54	53	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝑁‘(𝐴 − 𝑦)) = (𝑁‘(𝐴 − (𝑤 + 𝑥))))
55	54	breq2d 5112	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 + 𝑥) → ((𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − (𝑤 + 𝑥)))))
56		simplrr 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
57	26	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑊 ∈ LMod)
58	29	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑈 ∈ 𝐿)
59		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑈)
60	31	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑈)
61	34, 8	lssvacl 20906	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈)) → (𝑤 + 𝑥) ∈ 𝑈)
62	57, 58, 59, 60, 61	syl22anc 839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑤 + 𝑥) ∈ 𝑈)
63	55, 56, 62	rspcdva 3579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − (𝑤 + 𝑥))))
64		lmodgrp 20830	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
65	26, 64	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑊 ∈ Grp)
66	65	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑊 ∈ Grp)
67	33	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝐴 ∈ 𝑉)
68	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑉)
69	30	sselda 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑉)
70	1, 34, 2	grpsubsub4 18975	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
71	66, 67, 68, 69, 70	syl13anc 1375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
72	71	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘((𝐴 − 𝑥) − 𝑤)) = (𝑁‘(𝐴 − (𝑤 + 𝑥))))
73	63, 72	breqtrrd 5128	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘((𝐴 − 𝑥) − 𝑤)))
74	73	ralrimiva 3130	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → ∀𝑤 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘((𝐴 − 𝑥) − 𝑤)))
75	74	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘((𝐴 − 𝑥) − 𝑤)))
76		eqid 2737	. . . . . . . 8 ⊢ (((𝐴 − 𝑥) , 𝑧) / ((𝑧 , 𝑧) + 1)) = (((𝐴 − 𝑥) , 𝑧) / ((𝑧 , 𝑧) + 1))
77	1, 3, 34, 2, 48, 8, 49, 50, 51, 52, 75, 76	pjthlem1 25405	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → ((𝐴 − 𝑥) , 𝑧) = 0)
78	24	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ ℂPreHil)
79		cphclm 25157	. . . . . . . . 9 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
80	78, 79	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ ℂMod)
81		eqid 2737	. . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
82	81	clm0 25040	. . . . . . . 8 ⊢ (𝑊 ∈ ℂMod → 0 = (0_g‘(Scalar‘𝑊)))
83	80, 82	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → 0 = (0_g‘(Scalar‘𝑊)))
84	77, 83	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊)))
85	84	ralrimiva 3130	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → ∀𝑧 ∈ 𝑈 ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊)))
86		eqid 2737	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
87	1, 48, 81, 86, 42	elocv 21635	. . . . 5 ⊢ ((𝐴 − 𝑥) ∈ (𝑂‘𝑈) ↔ (𝑈 ⊆ 𝑉 ∧ (𝐴 − 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑈 ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊))))
88	30, 47, 85, 87	syl3anbrc 1345	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝐴 − 𝑥) ∈ (𝑂‘𝑈))
89		pjthlem.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝑊)
90	34, 89	lsmelvali 19591	. . . 4 ⊢ (((𝑈 ∈ (SubGrp‘𝑊) ∧ (𝑂‘𝑈) ∈ (SubGrp‘𝑊)) ∧ (𝑥 ∈ 𝑈 ∧ (𝐴 − 𝑥) ∈ (𝑂‘𝑈))) → (𝑥 + (𝐴 − 𝑥)) ∈ (𝑈 ⊕ (𝑂‘𝑈)))
91	39, 45, 31, 88, 90	syl22anc 839	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑥 + (𝐴 − 𝑥)) ∈ (𝑈 ⊕ (𝑂‘𝑈)))
92	36, 91	eqeltrrd 2838	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))
93	23, 92	rexlimddv 3145	1 ⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∃!wreu 3350 ⊆ wss 3903 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 + caddc 11041 ≤ cle 11179 / cdiv 11806 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 Scalarcsca 17192 ·_𝑖cip 17194 TopOpenctopn 17353 0_gc0g 17371 Grpcgrp 18875 -_gcsg 18877 SubGrpcsubg 19062 LSSumclsm 19575 Abelcabl 19722 LModclmod 20823 LSubSpclss 20894 PreHilcphl 21591 ocvcocv 21627 Clsdccld 22972 normcnm 24532 ℂModcclm 25030 ℂPreHilccph 25134 CMetSpccms 25300 ℂHilchl 25302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-subrng 20491 df-subrg 20515 df-drng 20676 df-staf 20784 df-srng 20785 df-lmod 20825 df-lss 20895 df-lmhm 20986 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-phl 21593 df-ocv 21630 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-flim 23895 df-fcls 23897 df-xms 24276 df-ms 24277 df-tms 24278 df-nm 24538 df-ngp 24539 df-nlm 24542 df-cncf 24839 df-clm 25031 df-cph 25136 df-cfil 25223 df-cmet 25225 df-cms 25303 df-bn 25304 df-hl 25305

This theorem is referenced by: pjth 25407

W3C validator