pjthlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pjthlem2		Structured version Visualization version GIF version

Theorem pjthlem2 24630

Description: Lemma for pjth 24631. (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 24556	. . . . 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 2844	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊))
10		pjthlem.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (Clsd‘𝐽))
11		hlcms 24558	. . . . . . 7 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp)
12	4, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CMetSp)
13	1, 8	lssss 20226	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ 𝑉)
14	7, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
15		eqid 2733	. . . . . . 7 ⊢ (𝑊 ↾_s 𝑈) = (𝑊 ↾_s 𝑈)
16		pjthlem.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑊)
17	15, 1, 16	cmsss 24543	. . . . . 6 ⊢ ((𝑊 ∈ CMetSp ∧ 𝑈 ⊆ 𝑉) → ((𝑊 ↾_s 𝑈) ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽)))
18	12, 14, 17	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((𝑊 ↾_s 𝑈) ∈ CMetSp ↔ 𝑈 ∈ (Clsd‘𝐽)))
19	10, 18	mpbird 256	. . . 4 ⊢ (𝜑 → (𝑊 ↾_s 𝑈) ∈ CMetSp)
20		pjthlem.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
21	1, 2, 3, 6, 9, 19, 20	minvec 24628	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))
22		reurex 3364	. . 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 24366	. . . . . 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 20198	. . . . 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 767	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑥 ∈ 𝑈)
32	30, 31	sseldd 3924	. . . 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 19446	. . . 4 ⊢ ((𝑊 ∈ Abel ∧ (𝑥 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → (𝑥 + (𝐴 − 𝑥)) = 𝐴)
36	28, 32, 33, 35	syl12anc 833	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑥 + (𝐴 − 𝑥)) = 𝐴)
37	8	lsssssubg 20248	. . . . . 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 3924	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝑈 ∈ (SubGrp‘𝑊))
40		cphphl 24363	. . . . . . 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 20905	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ⊆ 𝑉) → (𝑂‘𝑈) ∈ 𝐿)
44	41, 30, 43	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ 𝐿)
45	38, 44	sseldd 3924	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑂‘𝑈) ∈ (SubGrp‘𝑊))
46	1, 2	lmodvsubcl 20196	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) → (𝐴 − 𝑥) ∈ 𝑉)
47	26, 33, 32, 46	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝐴 − 𝑥) ∈ 𝑉)
48		pjthlem.h	. . . . . . . 8 ⊢ , = (·_𝑖‘𝑊)
49	4	ad2antrr 722	. . . . . . . 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 7303	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝐴 − 𝑦) = (𝐴 − (𝑤 + 𝑥)))
54	53	fveq2d 6796	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑤 + 𝑥) → (𝑁‘(𝐴 − 𝑦)) = (𝑁‘(𝐴 − (𝑤 + 𝑥))))
55	54	breq2d 5089	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑤 + 𝑥) → ((𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)) ↔ (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − (𝑤 + 𝑥)))))
56		simplrr 774	. . . . . . . . . . . 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 20244	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) ∧ (𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑈)) → (𝑤 + 𝑥) ∈ 𝑈)
62	57, 58, 59, 60, 61	syl22anc 835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑤 + 𝑥) ∈ 𝑈)
63	55, 56, 62	rspcdva 3564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − (𝑤 + 𝑥))))
64		lmodgrp 20158	. . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑉)
70	1, 34, 2	grpsubsub4 18696	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Grp ∧ (𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
71	66, 67, 68, 69, 70	syl13anc 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → ((𝐴 − 𝑥) − 𝑤) = (𝐴 − (𝑤 + 𝑥)))
72	71	fveq2d 6796	. . . . . . . . . . 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 5105	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑤 ∈ 𝑈) → (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘((𝐴 − 𝑥) − 𝑤)))
74	73	ralrimiva 3137	. . . . . . . . 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 2733	. . . . . . . 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 24629	. . . . . . 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 24381	. . . . . . . . 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 2733	. . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
82	81	clm0 24263	. . . . . . . 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 2773	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) ∧ 𝑧 ∈ 𝑈) → ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊)))
85	84	ralrimiva 3137	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → ∀𝑧 ∈ 𝑈 ((𝐴 − 𝑥) , 𝑧) = (0_g‘(Scalar‘𝑊)))
86		eqid 2733	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
87	1, 48, 81, 86, 42	elocv 20901	. . . . 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 1341	. . . 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 19283	. . . 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 835	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → (𝑥 + (𝐴 − 𝑥)) ∈ (𝑈 ⊕ (𝑂‘𝑈)))
92	36, 91	eqeltrrd 2835	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))) → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))
93	23, 92	rexlimddv 3152	1 ⊢ (𝜑 → 𝐴 ∈ (𝑈 ⊕ (𝑂‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ∃wrex 3068 ∃!wreu 3219 ⊆ wss 3889 class class class wbr 5077 ‘cfv 6447 (class class class)co 7295 0cc0 10899 1c1 10900 + caddc 10902 ≤ cle 11038 / cdiv 11660 Basecbs 16940 ↾_s cress 16969 +_gcplusg 16990 Scalarcsca 16993 ·_𝑖cip 16995 TopOpenctopn 17160 0_gc0g 17178 Grpcgrp 18605 -_gcsg 18607 SubGrpcsubg 18777 LSSumclsm 19267 Abelcabl 19415 LModclmod 20151 LSubSpclss 20221 PreHilcphl 20857 ocvcocv 20893 Clsdccld 22195 normcnm 23760 ℂModcclm 24253 ℂPreHilccph 24358 CMetSpccms 24524 ℂHilchl 24526

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 ax-addf 10978 ax-mulf 10979

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-tpos 8062 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-2o 8318 df-er 8518 df-map 8637 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-fi 9198 df-sup 9229 df-inf 9230 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-q 12717 df-rp 12759 df-xneg 12876 df-xadd 12877 df-xmul 12878 df-ioo 13111 df-ico 13113 df-icc 13114 df-fz 13268 df-fzo 13411 df-seq 13750 df-exp 13811 df-hash 14073 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-hom 17014 df-cco 17015 df-rest 17161 df-topn 17162 df-0g 17180 df-gsum 17181 df-topgen 17182 df-pt 17183 df-prds 17186 df-xrs 17241 df-qtop 17246 df-imas 17247 df-xps 17249 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-lsm 19269 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-ring 19813 df-cring 19814 df-oppr 19890 df-dvdsr 19911 df-unit 19912 df-invr 19942 df-dvr 19953 df-rnghom 19987 df-drng 20021 df-subrg 20050 df-staf 20133 df-srng 20134 df-lmod 20153 df-lss 20222 df-lmhm 20312 df-lvec 20393 df-sra 20462 df-rgmod 20463 df-psmet 20617 df-xmet 20618 df-met 20619 df-bl 20620 df-mopn 20621 df-fbas 20622 df-fg 20623 df-cnfld 20626 df-phl 20859 df-ocv 20896 df-top 22071 df-topon 22088 df-topsp 22110 df-bases 22124 df-cld 22198 df-ntr 22199 df-cls 22200 df-nei 22277 df-cn 22406 df-cnp 22407 df-haus 22494 df-cmp 22566 df-tx 22741 df-hmeo 22934 df-fil 23025 df-flim 23118 df-fcls 23120 df-xms 23501 df-ms 23502 df-tms 23503 df-nm 23766 df-ngp 23767 df-nlm 23770 df-cncf 24069 df-clm 24254 df-cph 24360 df-cfil 24447 df-cmet 24449 df-cms 24527 df-bn 24528 df-hl 24529

This theorem is referenced by: pjth 24631

W3C validator