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Theorem pjthlem2 25472

Description: Lemma for pjth 25473. (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 25398	. . . . 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 2851	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊))
10		pjthlem.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽))
11		hlcms 25400	. . . . . . 7 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
12	4, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CMetSp)
13	1, 8	lssss 20934	. . . . . . 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 25385	. . . . . 6 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑈 ⊆ 𝑉) → ((𝑊 ↾_s 𝑈) ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽)))
18	12, 14, 17	syl2anc 584	. . . . 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 25470	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
22		reurex 3384	. . 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 25208	. . . . . 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 20907	. . . . 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 3984	. . . 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 19834	. . . 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 20956	. . . . . 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 3984	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑈 ∈ (SubGrp‘𝑊))
40		cphphl 25205	. . . . . . 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 21690	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ⊆ 𝑉) → (𝑂‘𝑈) ∈ 𝐿)
44	41, 30, 43	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ 𝐿)
45	38, 44	sseldd 3984	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ (SubGrp‘𝑊))
46	1, 2	lmodvsubcl 20905	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝐴 − 𝑥) ∈ 𝑉)
47	26, 33, 32, 46	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝐴 − 𝑥) ∈ 𝑉)
48		pjthlem.h	. . . . . . . 8 ⊢ , = (·_𝑖‘𝑊)
49	4	ad2antrr 726	. . . . . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝐴 − 𝑦) = (𝐴 − (𝑤 + 𝑥)))
54	53	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝑁‘(𝐴 − 𝑦)) = (𝑁‘(𝐴 − (𝑤 + 𝑥))))
55	54	breq2d 5155	. . . . . . . . . . . 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 20941	. . . . . . . . . . . . 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 3623	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − (𝑤 + 𝑥))))
64		lmodgrp 20865	. . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑉)
70	1, 34, 2	grpsubsub4 19051	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
71	66, 67, 68, 69, 70	syl13anc 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
72	71	fveq2d 6910	. . . . . . . . . . 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 5171	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘((𝐴 − 𝑥) − 𝑤)))
74	73	ralrimiva 3146	. . . . . . . . 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 25471	. . . . . . 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 25223	. . . . . . . . 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 25105	. . . . . . . 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 2777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊)))
85	84	ralrimiva 3146	. . . . 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 21686	. . . . 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 1344	. . . 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 19668	. . . 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 2842	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))
93	23, 92	rexlimddv 3161	1 ⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 / cdiv 11920 Basecbs 17247 ↾_s cress 17274 +_gcplusg 17297 Scalarcsca 17300 ·_𝑖cip 17302 TopOpenctopn 17466 0_gc0g 17484 Grpcgrp 18951 -_gcsg 18953 SubGrpcsubg 19138 LSSumclsm 19652 Abelcabl 19799 LModclmod 20858 LSubSpclss 20929 PreHilcphl 21642 ocvcocv 21678 Clsdccld 23024 normcnm 24589 ℂModcclm 25095 ℂPreHilccph 25200 CMetSpccms 25366 ℂHilchl 25368

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-phl 21644 df-ocv 21681 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-flim 23947 df-fcls 23949 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nlm 24599 df-cncf 24904 df-clm 25096 df-cph 25202 df-cfil 25289 df-cmet 25291 df-cms 25369 df-bn 25370 df-hl 25371

This theorem is referenced by: pjth 25473

W3C validator