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Theorem nmlno0lem 31085

Description: Lemma for nmlno0i 31086. (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 30953	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝐾‘𝑥) ∈ ℝ)
5	1, 4	mpan 702	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → (𝐾‘𝑥) ∈ ℝ)
6	5	recnd 11236	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → (𝐾‘𝑥) ∈ ℂ)
7	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘𝑥) ∈ ℂ)
8		nmlno0lem.p	. . . . . . . . . . . . . . . . 17 ⊢ 𝑃 = (0_vec‘𝑈)
9	2, 8, 3	nvz 30961	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → ((𝐾‘𝑥) = 0 ↔ 𝑥 = 𝑃))
10	1, 9	mpan 702	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 → ((𝐾‘𝑥) = 0 ↔ 𝑥 = 𝑃))
11		fveq2 6882	. . . . . . . . . . . . . . . 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 31048	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → (𝑇‘𝑃) = 𝑄)
18	1, 12, 13, 17	mp3an 1487	. . . . . . . . . . . . . . . 16 ⊢ (𝑇‘𝑃) = 𝑄
19	11, 18	eqtrdi 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝑇‘𝑥) = 𝑄)
20	10, 19	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 → ((𝐾‘𝑥) = 0 → (𝑇‘𝑥) = 𝑄))
21	20	necon3d 2985	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 → ((𝑇‘𝑥) ≠ 𝑄 → (𝐾‘𝑥) ≠ 0))
22	21	imp 411	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘𝑥) ≠ 0)
23	7, 22	recne0d 11984	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (1 / (𝐾‘𝑥)) ≠ 0)
24		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘𝑥) ≠ 𝑄)
25	7, 22	reccld 11983	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (1 / (𝐾‘𝑥)) ∈ ℂ)
26	2, 14, 16	lnof 31047	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:𝑋⟶𝑌)
27	1, 12, 13, 26	mp3an 1487	. . . . . . . . . . . . . . . 16 ⊢ 𝑇:𝑋⟶𝑌
28	27	ffvelcdmi 7079	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 → (𝑇‘𝑥) ∈ 𝑌)
29	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘𝑥) ∈ 𝑌)
30		nmlno0lem.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑊)
31	14, 30, 15	nvmul0or 30942	. . . . . . . . . . . . . . 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 1474	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = 𝑄 ↔ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
34	33	necon3abid 3000	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄 ↔ ¬ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄)))
35		neanior 3057	. . . . . . . . . . . 12 ⊢ (((1 / (𝐾‘𝑥)) ≠ 0 ∧ (𝑇‘𝑥) ≠ 𝑄) ↔ ¬ ((1 / (𝐾‘𝑥)) = 0 ∨ (𝑇‘𝑥) = 𝑄))
36	34, 35	bitr4di 292	. . . . . . . . . . 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 725	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ≠ 𝑄)
38	14, 30	nvscl 30918	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ NrmCVec ∧ (1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌)
39	12, 38	mp3an1 1474	. . . . . . . . . . . 12 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ (𝑇‘𝑥) ∈ 𝑌) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌)
40	25, 29, 39	syl2anc 595	. . . . . . . . . . 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 30966	. . . . . . . . . . 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 598	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))))
45	44	ex 417	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝑇‘𝑥) ≠ 𝑄 → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
46	45	adantl 486	. . . . . . 7 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ((𝑇‘𝑥) ≠ 𝑄 → 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
47	14, 41	nmosetre 31056	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ)
48	12, 27, 47	mp2an 704	. . . . . . . . . . . . 13 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ
49		ressxr 11252	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
50	48, 49	sstri 3954	. . . . . . . . . . . 12 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))} ⊆ ℝ^*
51		simpl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → 𝑥 ∈ 𝑋)
52		nmlno0lem.r	. . . . . . . . . . . . . . . . 17 ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑈)
53	2, 52	nvscl 30918	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ NrmCVec ∧ (1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
54	1, 53	mp3an1 1474	. . . . . . . . . . . . . . 15 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
55	25, 51, 54	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑅𝑥) ∈ 𝑋)
56	19	necon3i 2996	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇‘𝑥) ≠ 𝑄 → 𝑥 ≠ 𝑃)
57	2, 52, 8, 3	nv1 30967	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑥 ≠ 𝑃) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
58	1, 57	mp3an1 1474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑥 ≠ 𝑃) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
59	56, 58	sylan2 604	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1)
60		1re 11207	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
61	59, 60	eqeltrdi 2877	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ∈ ℝ)
62		eqle 11311	. . . . . . . . . . . . . . 15 ⊢ (((𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ∈ ℝ ∧ (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) = 1) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1)
63	61, 59, 62	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1)
64	1, 12, 13	3pm3.2i 1356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿)
65	2, 52, 30, 16	lnomul 31052	. . . . . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . . . . 17 ⊢ (((1 / (𝐾‘𝑥)) ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)) = ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))
67	25, 51, 66	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)) = ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))
68	67	eqcomd 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) = (𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥)))
69	68	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))
70		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → (𝐾‘𝑧) = (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)))
71	70	breq1d 5123	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → ((𝐾‘𝑧) ≤ 1 ↔ (𝐾‘((1 / (𝐾‘𝑥))𝑅𝑥)) ≤ 1))
72		2fveq3 6887	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ((1 / (𝐾‘𝑥))𝑅𝑥) → (𝑀‘(𝑇‘𝑧)) = (𝑀‘(𝑇‘((1 / (𝐾‘𝑥))𝑅𝑥))))
73	72	eqeq2d 2780	. . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . 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 849	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) = (𝑀‘(𝑇‘𝑧))))
77		fvex 6895	. . . . . . . . . . . . . 14 ⊢ (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ V
78		eqeq1 2773	. . . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . . . 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 3195	. . . . . . . . . . . . . 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 3647	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 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 13349	. . . . . . . . . . . 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 598	. . . . . . . . . . 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 726	. . . . . . . . . 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 31055	. . . . . . . . . . . . . 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 1487	. . . . . . . . . . . . 13 ⊢ (𝑁‘𝑇) = sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < )
89	88	eqeq1i 2774	. . . . . . . . . . . 12 ⊢ ((𝑁‘𝑇) = 0 ↔ sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ) = 0)
90	89	biimpi 219	. . . . . . . . . . 11 ⊢ ((𝑁‘𝑇) = 0 → sup({𝑦 ∣ ∃𝑧 ∈ 𝑋 ((𝐾‘𝑧) ≤ 1 ∧ 𝑦 = (𝑀‘(𝑇‘𝑧)))}, ℝ^*, < ) = 0)
91	90	ad2antrr 738	. . . . . . . . . 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 5141	. . . . . . . . 9 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ≤ 0)
93	14, 41	nvcl 30953	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ NrmCVec ∧ ((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)) ∈ 𝑌) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ ℝ)
94	12, 40, 93	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑇‘𝑥) ≠ 𝑄) → (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))) ∈ ℝ)
95		0re 11209	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
96		lenlt 11287	. . . . . . . . . . 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 597	. . . . . . . . . 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 726	. . . . . . . . 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 235	. . . . . . . 8 ⊢ ((((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑇‘𝑥) ≠ 𝑄) → ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥))))
100	99	ex 417	. . . . . . 7 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ((𝑇‘𝑥) ≠ 𝑄 → ¬ 0 < (𝑀‘((1 / (𝐾‘𝑥))𝑆(𝑇‘𝑥)))))
101	46, 100	pm2.65d 199	. . . . . 6 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → ¬ (𝑇‘𝑥) ≠ 𝑄)
102		nne 2968	. . . . . 6 ⊢ (¬ (𝑇‘𝑥) ≠ 𝑄 ↔ (𝑇‘𝑥) = 𝑄)
103	101, 102	sylib 221	. . . . 5 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑇‘𝑥) = 𝑄)
104		nmlno0.0	. . . . . . . 8 ⊢ 𝑍 = (𝑈 0_op 𝑊)
105	2, 15, 104	0oval 31080	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (𝑍‘𝑥) = 𝑄)
106	1, 12, 105	mp3an12 1477	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 → (𝑍‘𝑥) = 𝑄)
107	106	adantl 486	. . . . 5 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑍‘𝑥) = 𝑄)
108	103, 107	eqtr4d 2807	. . . 4 ⊢ (((𝑁‘𝑇) = 0 ∧ 𝑥 ∈ 𝑋) → (𝑇‘𝑥) = (𝑍‘𝑥))
109	108	ralrimiva 3163	. . 3 ⊢ ((𝑁‘𝑇) = 0 → ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥))
110		ffn 6706	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → 𝑇 Fn 𝑋)
111	27, 110	ax-mp 5	. . . 4 ⊢ 𝑇 Fn 𝑋
112	2, 14, 104	0oo 31081	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌)
113	1, 12, 112	mp2an 704	. . . . 5 ⊢ 𝑍:𝑋⟶𝑌
114		ffn 6706	. . . . 5 ⊢ (𝑍:𝑋⟶𝑌 → 𝑍 Fn 𝑋)
115	113, 114	ax-mp 5	. . . 4 ⊢ 𝑍 Fn 𝑋
116		eqfnfv 7026	. . . 4 ⊢ ((𝑇 Fn 𝑋 ∧ 𝑍 Fn 𝑋) → (𝑇 = 𝑍 ↔ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥)))
117	111, 115, 116	mp2an 704	. . 3 ⊢ (𝑇 = 𝑍 ↔ ∀𝑥 ∈ 𝑋 (𝑇‘𝑥) = (𝑍‘𝑥))
118	109, 117	sylibr 237	. 2 ⊢ ((𝑁‘𝑇) = 0 → 𝑇 = 𝑍)
119		fveq2 6882	. . 3 ⊢ (𝑇 = 𝑍 → (𝑁‘𝑇) = (𝑁‘𝑍))
120	86, 104	nmoo0 31083	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁‘𝑍) = 0)
121	1, 12, 120	mp2an 704	. . 3 ⊢ (𝑁‘𝑍) = 0
122	119, 121	eqtrdi 2820	. 2 ⊢ (𝑇 = 𝑍 → (𝑁‘𝑇) = 0)
123	118, 122	impbii 212	1 ⊢ ((𝑁‘𝑇) = 0 ↔ 𝑇 = 𝑍)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 {cab 2747 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 class class class wbr 5113 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 supcsup 9399 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 ℝ^*cxr 11241 < clt 11242 ≤ cle 11243 / cdiv 11870 NrmCVeccnv 30876 BaseSetcba 30878 ·_𝑠OLD cns 30879 0_veccn0v 30880 norm_CVcnmcv 30882 LnOp clno 31032 normOp_OLD cnmoo 31033 0_op c0o 31035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9401 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-grpo 30785 df-gid 30786 df-ginv 30787 df-ablo 30837 df-vc 30851 df-nv 30884 df-va 30887 df-ba 30888 df-sm 30889 df-0v 30890 df-nmcv 30892 df-lno 31036 df-nmoo 31037 df-0o 31039

This theorem is referenced by: nmlno0i 31086

W3C validator