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Theorem nmlno0lem 28564

Description: Lemma for nmlno0i 28565. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nmlno0.3	⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
nmlno0.0	⊢ 𝑍 = (𝑈 0_op 𝑊)
nmlno0.7	⊢ 𝐿 = (𝑈 LnOp 𝑊)
nmlno0lem.u	⊢ 𝑈 ∈ NrmCVec
nmlno0lem.w	⊢ 𝑊 ∈ NrmCVec
nmlno0lem.l	⊢ 𝑇 ∈ 𝐿
nmlno0lem.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmlno0lem.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmlno0lem.r	⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈)
nmlno0lem.s	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊)
nmlno0lem.p	⊢ 𝑃 = (0_vec‘𝑈)
nmlno0lem.q	⊢ 𝑄 = (0_vec‘𝑊)
nmlno0lem.k	⊢ 𝐾 = (norm_CV‘𝑈)
nmlno0lem.m	⊢ 𝑀 = (norm_CV‘𝑊)

**Assertion**
Ref	Expression
nmlno0lem	⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)

Proof of Theorem nmlno0lem Dummy variables 𝑦 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmlno0lem.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 ∈ NrmCVec
2		nmlno0lem.1	. . . . . . . . . . . . . . . 16 ⊢ 𝑋 = (BaseSet‘𝑈)
3		nmlno0lem.k	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = (norm_CV‘𝑈)
4	2, 3	nvcl 28432	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐾‘𝑥) ∈ ℝ)
5	1, 4	mpan 688	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → (𝐾‘𝑥) ∈ ℝ)
6	5	recnd 10663	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝐾‘𝑥) ∈ ℂ)
7	6	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘𝑥) ∈ ℂ)
8		nmlno0lem.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (0_vec‘𝑈)
9	2, 8, 3	nvz 28440	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → ((𝐾‘𝑥) = 0 ↔ 𝑥 = 𝑃))
10	1, 9	mpan 688	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 → ((𝐾‘𝑥) = 0 ↔ 𝑥 = 𝑃))
11		fveq2 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑃 → (𝑇‘𝑥) = (𝑇‘𝑃))
12		nmlno0lem.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 ∈ NrmCVec
13		nmlno0lem.l	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 ∈ 𝐿
14		nmlno0lem.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑌 = (BaseSet‘𝑊)
15		nmlno0lem.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (0_vec‘𝑊)
16		nmlno0.7	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐿 = (𝑈 LnOp 𝑊)
17	2, 14, 8, 15, 16	lno0 28527	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑃) = 𝑄)
18	1, 12, 13, 17	mp3an 1457	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘𝑃) = 𝑄
19	11, 18	syl6eq 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝑇‘𝑥) = 𝑄)
20	10, 19	syl6bi 255	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝐾‘𝑥) = 0 → (𝑇‘𝑥) = 𝑄))
21	20	necon3d 3037	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝑇‘𝑥) ≠ 𝑄 → (𝐾‘𝑥) ≠ 0))
22	21	imp 409	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘𝑥) ≠ 0)
23	7, 22	recne0d 11404	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (1 / (𝐾‘𝑥)) ≠ 0)
24		simpr 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘𝑥) ≠ 𝑄)
25	7, 22	reccld 11403	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (1 / (𝐾‘𝑥)) ∈ ℂ)
26	2, 14, 16	lnof 28526	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)
27	1, 12, 13, 26	mp3an 1457	. . . . . . . . . . . . . . . 16 ⊢ 𝑇:𝑋⟶𝑌
28	27	ffvelrni 6845	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 → (𝑇‘𝑥) ∈ 𝑌)
29	28	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘𝑥) ∈ 𝑌)
30		nmlno0lem.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊)
31	14, 30, 15	nvmul0or 28421	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ NrmCVec ∧ (1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = 𝑄 ↔ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
32	12, 31	mp3an1 1444	. . . . . . . . . . . . . 14 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = 𝑄 ↔ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
33	25, 29, 32	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = 𝑄 ↔ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
34	33	necon3abid 3052	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄 ↔ ¬ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
35		neanior 3109	. . . . . . . . . . . 12 ⊢ (((1 / (𝐾‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 𝑄) ↔ ¬ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄))
36	34, 35	syl6bbr 291	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄 ↔ ((1 / (𝐾‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 𝑄)))
37	23, 24, 36	mpbir2and 711	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄)
38	14, 30	nvscl 28397	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌)
39	12, 38	mp3an1 1444	. . . . . . . . . . . 12 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌)
40	25, 29, 39	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌)
41		nmlno0lem.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (norm_CV‘𝑊)
42	14, 15, 41	nvgt0 28445	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ NrmCVec ∧ ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄 ↔ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
43	12, 40, 42	sylancr 589	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄 ↔ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
44	37, 43	mpbid 234	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))))
45	44	ex 415	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝑇‘𝑥) ≠ 𝑄 → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
46	45	adantl 484	. . . . . . 7 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ((𝑇‘𝑥) ≠ 𝑄 → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
47	14, 41	nmosetre 28535	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ)
48	12, 27, 47	mp2an 690	. . . . . . . . . . . . 13 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ
49		ressxr 10679	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
50	48, 49	sstri 3976	. . . . . . . . . . . 12 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ^*
51		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → 𝑥 ∈ 𝑋)
52		nmlno0lem.r	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈)
53	2, 52	nvscl 28397	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
54	1, 53	mp3an1 1444	. . . . . . . . . . . . . . 15 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
55	25, 51, 54	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
56	19	necon3i 3048	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 𝑄 → 𝑥 ≠ 𝑃)
57	2, 52, 8, 3	nv1 28446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 ≠ 𝑃) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
58	1, 57	mp3an1 1444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ≠ 𝑃) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
59	56, 58	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
60		1re 10635	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
61	59, 60	eqeltrdi 2921	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ∈ ℝ)
62		eqle 10736	. . . . . . . . . . . . . . 15 ⊢ (((𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ∈ ℝ ∧ (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1)
63	61, 59, 62	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1)
64	1, 12, 13	3pm3.2i 1335	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿)
65	2, 52, 30, 16	lnomul 28531	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ ((1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋)) → (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)) = ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))
66	64, 65	mpan 688	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)) = ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))
67	25, 51, 66	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)) = ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))
68	67	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)))
69	68	fveq2d 6669	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))
70		fveq2 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → (𝐾‘𝑧) = (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)))
71	70	breq1d 5069	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → ((𝐾‘𝑧) ≤ 1 ↔ (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1))
72		2fveq3 6670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → (𝑀‘(𝑇‘𝑧)) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))
73	72	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → ((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧)) ↔ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)))))
74	71, 73	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → (((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))) ↔ ((𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))))
75	74	rspcev 3623	. . . . . . . . . . . . . 14 ⊢ ((((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋 ∧ ((𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))) → ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))))
76	55, 63, 69, 75	syl12anc 834	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))))
77		fvex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ V
78		eqeq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) → (𝑦 = (𝑀‘(𝑇‘𝑧)) ↔ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))))
79	78	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) → (((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧))) ↔ ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧)))))
80	79	rexbidv 3297	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) → (∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧)))))
81	77, 80	elab 3667	. . . . . . . . . . . . 13 ⊢ ((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ↔ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))))
82	76, 81	sylibr 236	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))})
83		supxrub 12711	. . . . . . . . . . . 12 ⊢ (({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ^* ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))
84	50, 82, 83	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))
85	84	adantll 712	. . . . . . . . . 10 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))
86		nmlno0.3	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (𝑈 normOp_OLD 𝑊)
87	2, 14, 3, 41, 86	nmooval 28534	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ))
88	1, 12, 27, 87	mp3an 1457	. . . . . . . . . . . . 13 ⊢ (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < )
89	88	eqeq1i 2826	. . . . . . . . . . . 12 ⊢ ((𝑁‘𝑇) = 0 ↔ sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ) = 0)
90	89	biimpi 218	. . . . . . . . . . 11 ⊢ ((𝑁‘𝑇) = 0 → sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ) = 0)
91	90	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ) = 0)
92	85, 91	breqtrd 5085	. . . . . . . . 9 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ 0)
93	14, 41	nvcl 28432	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ ℝ)
94	12, 40, 93	sylancr 589	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ ℝ)
95		0re 10637	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
96		lenlt 10713	. . . . . . . . . . 11 ⊢ (((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ 0 ↔ ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
97	94, 95, 96	sylancl 588	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ 0 ↔ ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
98	97	adantll 712	. . . . . . . . 9 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → ((𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ 0 ↔ ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
99	92, 98	mpbid 234	. . . . . . . 8 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))))
100	99	ex 415	. . . . . . 7 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ((𝑇‘𝑥) ≠ 𝑄 → ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
101	46, 100	pm2.65d 198	. . . . . 6 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ¬ (𝑇‘𝑥) ≠ 𝑄)
102		nne 3020	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 𝑄 ↔ (𝑇‘𝑥) = 𝑄)
103	101, 102	sylib 220	. . . . 5 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑇‘𝑥) = 𝑄)
104		nmlno0.0	. . . . . . . 8 ⊢ 𝑍 = (𝑈 0_op 𝑊)
105	2, 15, 104	0oval 28559	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑍‘𝑥) = 𝑄)
106	1, 12, 105	mp3an12 1447	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑍‘𝑥) = 𝑄)
107	106	adantl 484	. . . . 5 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑍‘𝑥) = 𝑄)
108	103, 107	eqtr4d 2859	. . . 4 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑇‘𝑥) = (𝑍‘𝑥))
109	108	ralrimiva 3182	. . 3 ⊢ ((𝑁‘𝑇) = 0 → ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥))
110		ffn 6509	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → 𝑇 Fn 𝑋)
111	27, 110	ax-mp 5	. . . 4 ⊢ 𝑇 Fn 𝑋
112	2, 14, 104	0oo 28560	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌)
113	1, 12, 112	mp2an 690	. . . . 5 ⊢ 𝑍:𝑋⟶𝑌
114		ffn 6509	. . . . 5 ⊢ (𝑍:𝑋⟶𝑌 → 𝑍 Fn 𝑋)
115	113, 114	ax-mp 5	. . . 4 ⊢ 𝑍 Fn 𝑋
116		eqfnfv 6797	. . . 4 ⊢ ((𝑇 Fn 𝑋 ∧ 𝑍 Fn 𝑋) → (𝑇 = 𝑍 ↔ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥)))
117	111, 115, 116	mp2an 690	. . 3 ⊢ (𝑇 = 𝑍 ↔ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥))
118	109, 117	sylibr 236	. 2 ⊢ ((𝑁‘𝑇) = 0 → 𝑇 = 𝑍)
119		fveq2 6665	. . 3 ⊢ (𝑇 = 𝑍 → (𝑁‘𝑇) = (𝑁‘𝑍))
120	86, 104	nmoo0 28562	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0)
121	1, 12, 120	mp2an 690	. . 3 ⊢ (𝑁‘𝑍) = 0
122	119, 121	syl6eq 2872	. 2 ⊢ (𝑇 = 𝑍 → (𝑁‘𝑇) = 0)
123	118, 122	impbii 211	1 ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5059 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 supcsup 8898 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 / cdiv 11291 NrmCVeccnv 28355 BaseSetcba 28357 ·_𝑠OLD cns 28358 0_veccn0v 28359 norm_CVcnmcv 28361 LnOp clno 28511 normOp_OLD cnmoo 28512 0_op c0o 28514

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-lno 28515 df-nmoo 28516 df-0o 28518

This theorem is referenced by: nmlno0i 28565

W3C validator