hoicvr - Mathbox for Glauco Siliprandi

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Theorem hoicvr 43976

Description: 𝐼 is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
hoicvr.2	⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
hoicvr.3	⊢ (𝜑 → 𝑋 ∈ Fin)

**Assertion**
Ref	Expression
hoicvr	⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))

Distinct variable groups: 𝑖,𝑋,𝑗,𝑥 𝜑,𝑖,𝑗,𝑥 Allowed substitution hints: 𝐼(𝑥,𝑖,𝑗)

Proof of Theorem hoicvr Dummy variables 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10893	. . . . . . 7 ⊢ ℝ ∈ V
2		mapdm0 8588	. . . . . . 7 ⊢ (ℝ ∈ V → (ℝ ↑_m ∅) = {∅})
3	1, 2	ax-mp 5	. . . . . 6 ⊢ (ℝ ↑_m ∅) = {∅}
4	3	a1i 11	. . . . 5 ⊢ (𝑋 = ∅ → (ℝ ↑_m ∅) = {∅})
5		oveq2 7263	. . . . 5 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
6		ixpeq1 8654	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
7	6	iuneq2d 4950	. . . . . 6 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
8		ixp0x 8672	. . . . . . . . . 10 ⊢ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
10	9	iuneq2i 4942	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ {∅}
11		1nn 11914	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
12	11	ne0ii 4268	. . . . . . . . 9 ⊢ ℕ ≠ ∅
13		iunconst 4930	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ {∅} = {∅}
15	10, 14	eqtri 2766	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
16	15	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
17	7, 16	eqtrd 2778	. . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
18	4, 5, 17	3eqtr4d 2788	. . . 4 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
19		eqimss 3973	. . . 4 ⊢ ((ℝ ↑_m 𝑋) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
20	18, 19	syl 17	. . 3 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
21	20	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
22		simpll 763	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝜑)
23		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓 ∈ (ℝ ↑_m 𝑋))
24		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ¬ 𝑋 = ∅)
25		rncoss 5870	. . . . . . . . . . 11 ⊢ ran (abs ∘ 𝑓) ⊆ ran abs
26		absf 14977	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
27		frn 6591	. . . . . . . . . . . 12 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
28	26, 27	ax-mp 5	. . . . . . . . . . 11 ⊢ ran abs ⊆ ℝ
29	25, 28	sstri 3926	. . . . . . . . . 10 ⊢ ran (abs ∘ 𝑓) ⊆ ℝ
30	29	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ⊆ ℝ)
31		ltso 10986	. . . . . . . . . . 11 ⊢ < Or ℝ
32	31	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → < Or ℝ)
33	26	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → abs:ℂ⟶ℝ)
34		elmapi 8595	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → 𝑓:𝑋⟶ℝ)
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓:𝑋⟶ℝ)
36		ax-resscn 10859	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ℝ ⊆ ℂ)
38	35, 37	fssd 6602	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓:𝑋⟶ℂ)
39		fco 6608	. . . . . . . . . . . . 13 ⊢ ((abs:ℂ⟶ℝ ∧ 𝑓:𝑋⟶ℂ) → (abs ∘ 𝑓):𝑋⟶ℝ)
40	33, 38, 39	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → (abs ∘ 𝑓):𝑋⟶ℝ)
41		hoicvr.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ Fin)
42	41	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
43		rnffi 42600	. . . . . . . . . . . 12 ⊢ (((abs ∘ 𝑓):𝑋⟶ℝ ∧ 𝑋 ∈ Fin) → ran (abs ∘ 𝑓) ∈ Fin)
44	40, 42, 43	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ran (abs ∘ 𝑓) ∈ Fin)
45	44	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ∈ Fin)
46	34	frnd 6592	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → ran 𝑓 ⊆ ℝ)
47	26	fdmi 6596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom abs = ℂ
48	47	eqcomi 2747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = dom abs
49	36, 48	sseqtri 3953	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ dom abs
50	49	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → ℝ ⊆ dom abs)
51	46, 50	sstrd 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → ran 𝑓 ⊆ dom abs)
52		dmcosseq 5871	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑓 ⊆ dom abs → dom (abs ∘ 𝑓) = dom 𝑓)
53	51, 52	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom (abs ∘ 𝑓) = dom 𝑓)
54	34	fdmd 6595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom 𝑓 = 𝑋)
55	53, 54	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom (abs ∘ 𝑓) = 𝑋)
56	55	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) = 𝑋)
57		neqne 2950	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
58	57	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
59	56, 58	eqnetrd 3010	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) ≠ ∅)
60	59	neneqd 2947	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ dom (abs ∘ 𝑓) = ∅)
61		dm0rn0 5823	. . . . . . . . . . . . 13 ⊢ (dom (abs ∘ 𝑓) = ∅ ↔ ran (abs ∘ 𝑓) = ∅)
62	60, 61	sylnib 327	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ ran (abs ∘ 𝑓) = ∅)
63	62	neqned 2949	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
64	63	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
65		fisupcl 9158	. . . . . . . . . 10 ⊢ (( < Or ℝ ∧ (ran (abs ∘ 𝑓) ∈ Fin ∧ ran (abs ∘ 𝑓) ≠ ∅ ∧ ran (abs ∘ 𝑓) ⊆ ℝ)) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
66	32, 45, 64, 30, 65	syl13anc 1370	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
67	30, 66	sseldd 3918	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
68		arch 12160	. . . . . . . 8 ⊢ (sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ → ∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
69	67, 68	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
70	35	ffnd 6585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓 Fn 𝑋)
71	70	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 Fn 𝑋)
72	71	adantlr 711	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 Fn 𝑋)
73		simplll 771	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)))
74		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
75	74	ad3antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℕ)
76		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
77		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
78		simp2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑗 ∈ ℕ)
79		zssre 12256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ ⊆ ℝ
80		ressxr 10950	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℝ^*
81	79, 80	sstri 3926	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ⊆ ℝ^*
82		nnnegz 12252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℤ)
83	81, 82	sselid 3915	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ^*)
84	83	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ^*)
85	78, 84	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ^*)
86	74	nnxrd 42474	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ^*)
87	86	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ^*)
88	78, 87	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ^*)
89	34	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ)
90	80	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → ℝ ⊆ ℝ^*)
91	89, 90	fssd 6602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ^*)
92	91	3adant1l 1174	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ^*)
93	92	ffvelrnda 6943	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ^*)
94		nnre 11910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
95	94	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
96	95	3ad2antl2 1184	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
97	96	renegcld 11332	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ)
98	35	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
99	98	3ad2antl1 1183	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
100	99	renegcld 11332	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
101		simpll 763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝜑)
102		simplr 765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑_m 𝑋))
103		n0i 4264	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ 𝑋 → ¬ 𝑋 = ∅)
104	103	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ¬ 𝑋 = ∅)
105	101, 102, 104, 67	syl21anc 834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
106	105	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
107	34	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
108	36, 107	sselid 3915	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℂ)
109	108	abscld 15076	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
110	109	adantll 710	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
111	110	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
112	107	renegcld 11332	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
113	112	leabsd 15054	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘-(𝑓‘𝑖)))
114	108	absnegd 15089	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘-(𝑓‘𝑖)) = (abs‘(𝑓‘𝑖)))
115	113, 114	breqtrd 5096	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
116	115	adantll 710	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
117	116	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
118	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ⊆ ℝ)
119	104, 64	syldan 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
120	119	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
121		fimaxre2 11850	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ran (abs ∘ 𝑓) ⊆ ℝ ∧ ran (abs ∘ 𝑓) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
122	29, 44, 121	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
123	122	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
124	123	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
125		elmapfun 8612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → Fun 𝑓)
126	125	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → Fun 𝑓)
127		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
128	54	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → 𝑋 = dom 𝑓)
129	128	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑋 = dom 𝑓)
130	127, 129	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom 𝑓)
131		fvco 6848	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
132	126, 130, 131	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
133	132	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) = ((abs ∘ 𝑓)‘𝑖))
134		absfun 42779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ Fun abs
135	134	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → Fun abs)
136		funco 6458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun abs ∧ Fun 𝑓) → Fun (abs ∘ 𝑓))
137	135, 125, 136	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → Fun (abs ∘ 𝑓))
138	137	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → Fun (abs ∘ 𝑓))
139	108, 48	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ dom abs)
140		dmfco 6846	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
141	126, 130, 140	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
142	139, 141	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom (abs ∘ 𝑓))
143		fvelrn 6936	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun (abs ∘ 𝑓) ∧ 𝑖 ∈ dom (abs ∘ 𝑓)) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
144	138, 142, 143	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
145	133, 144	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
146	145	adantll 710	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
147	146	3ad2antl1 1183	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
148		suprub 11866	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ran (abs ∘ 𝑓) ⊆ ℝ ∧ ran (abs ∘ 𝑓) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦) ∧ (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓)) → (abs‘(𝑓‘𝑖)) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
149	118, 120, 124, 147, 148	syl31anc 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
150	100, 111, 106, 117, 149	letrd 11062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
151		simpl3 1191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
152	100, 106, 96, 150, 151	lelttrd 11063	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) < 𝑗)
153	100, 96	ltnegd 11483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (-(𝑓‘𝑖) < 𝑗 ↔ -𝑗 < --(𝑓‘𝑖)))
154	152, 153	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 < --(𝑓‘𝑖))
155	36, 99	sselid 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℂ)
156	155	negnegd 11253	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → --(𝑓‘𝑖) = (𝑓‘𝑖))
157	154, 156	breqtrd 5096	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 < (𝑓‘𝑖))
158	97, 99, 157	ltled 11053	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ≤ (𝑓‘𝑖))
159	99	leabsd 15054	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
160	99, 111, 106, 159, 149	letrd 11062	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
161	99, 106, 96, 160, 151	lelttrd 11063	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) < 𝑗)
162	85, 88, 93, 158, 161	elicod 13058	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
163	73, 75, 76, 77, 162	syl31anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
164	163	adantl3r 746	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
165		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
166		mptexg 7079	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ Fin → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
167	41, 166	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
168	167	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
169		hoicvr.2	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
170	169	fvmpt2 6868	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
171	165, 168, 170	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
172	171	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
173	172	3adant3 1130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
174		eqidd 2739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
175		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩
176	175	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑥 = 𝑖) → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
177		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ 𝑋)
178		opex 5373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
179	178	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → ⟨-𝑗, 𝑗⟩ ∈ V)
180	174, 176, 177, 179	fvmptd 6864	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
181	180	3ad2ant3 1133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
182	173, 181	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ⟨-𝑗, 𝑗⟩)
183	182	fveq2d 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1^st ‘((𝐼‘𝑗)‘𝑖)) = (1^st ‘⟨-𝑗, 𝑗⟩))
184		negex 11149	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝑗 ∈ V
185		vex 3426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑗 ∈ V
186	184, 185	op1st 7812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1^st ‘⟨-𝑗, 𝑗⟩) = -𝑗
187	186	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1^st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
188	183, 187	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1^st ‘((𝐼‘𝑗)‘𝑖)) = -𝑗)
189	182	fveq2d 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2^nd ‘((𝐼‘𝑗)‘𝑖)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
190	184, 185	op2nd 7813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2^nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
191	190	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2^nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
192	189, 191	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2^nd ‘((𝐼‘𝑗)‘𝑖)) = 𝑗)
193	188, 192	oveq12d 7273	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))) = (-𝑗[,)𝑗))
194	193	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
195	194	3adant1r 1175	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
196	195	ad5ant135 1366	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
197	164, 196	eleqtrd 2841	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
198	79, 82	sselid 3915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
199		opelxpi 5617	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
200	198, 94, 199	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
201	200	ad2antlr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
202		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
203	201, 202	fmptd 6970	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
204	171	feq1d 6569	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
205	203, 204	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
206	205	ad4ant14 748	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
207	206	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
208		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
209	207, 208	fvovco 42621	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑖) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
210	209	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))) = (([,) ∘ (𝐼‘𝑗))‘𝑖))
211	197, 210	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
212	211	ralrimiva 3107	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
213	72, 212	jca 511	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → (𝑓 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖)))
214		vex 3426	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
215	214	elixp 8650	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖)))
216	213, 215	sylibr 233	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
217	216	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) → (sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗 → 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)))
218	217	reximdva 3202	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → (∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗 → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)))
219	69, 218	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
220	22, 23, 24, 219	syl21anc 834	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
221		eliun 4925	. . . . 5 ⊢ (𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
222	220, 221	sylibr 233	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
223	222	ralrimiva 3107	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∀𝑓 ∈ (ℝ ↑_m 𝑋)𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
224		dfss3 3905	. . 3 ⊢ ((ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ ∀𝑓 ∈ (ℝ ↑_m 𝑋)𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
225	223, 224	sylibr 233	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
226	21, 225	pm2.61dan 809	1 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⊆ wss 3883 ∅c0 4253 {csn 4558 ⟨cop 4564 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 Or wor 5493 × cxp 5578 dom cdm 5580 ran crn 5581 ∘ ccom 5584 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 ↑_m cmap 8573 Xcixp 8643 Fincfn 8691 supcsup 9129 ℂcc 10800 ℝcr 10801 1c1 10803 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 -cneg 11136 ℕcn 11903 ℤcz 12249 [,)cico 13010 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: hoicvrrex 43984

W3C validator