hoicvr - Mathbox for Glauco Siliprandi

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Theorem hoicvr 47377

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.) Avoid ax-rep 5232 and shorten proof. (Revised by GG, 2-Apr-2026.)

**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 11216	. . . . . 6 ⊢ ℝ ∈ V
2		mapdm0 8842	. . . . . 6 ⊢ (ℝ ∈ V → (ℝ ↑_m ∅) = {∅})
3	1, 2	ax-mp 5	. . . . 5 ⊢ (ℝ ↑_m ∅) = {∅}
4		oveq2 7422	. . . . 5 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = (ℝ ↑_m ∅))
5		ixpeq1 8916	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
6	5	iuneq2d 4981	. . . . . 6 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
7		ixp0x 8934	. . . . . . . . 9 ⊢ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
9	8	iuneq2i 4973	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ {∅}
10		nnn0 46208	. . . . . . . 8 ⊢ ℕ ≠ ∅
11		iunconst 4961	. . . . . . . 8 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ {∅} = {∅}
13	9, 12	eqtri 2783	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
14	6, 13	eqtrdi 2811	. . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
15	3, 4, 14	3eqtr4a 2821	. . . 4 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
16	15	eqimssd 3987	. . 3 ⊢ (𝑋 = ∅ → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
17	16	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
18		elmapi 8849	. . . . . . . . . 10 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → 𝑓:𝑋⟶ℝ)
19	18	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓:𝑋⟶ℝ)
20	19	ffnd 6704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓 Fn 𝑋)
21	20	ad3antrrr 743	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 Fn 𝑋)
22		simplll 787	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)))
23		simpllr 788	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℕ)
24		simplr 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
25		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
26		nnnegz 12619	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℤ)
27	26	zxrd 46282	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ^*)
28	27	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ^*)
29	28	3ad2antl2 1205	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ^*)
30		nnxr 46109	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ^*)
31	30	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ^*)
32	31	3ad2antl2 1205	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ^*)
33	18	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ)
34	33	frexr 46215	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ^*)
35	34	3adant1l 1195	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ^*)
36	35	ffvelcdmda 7078	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ^*)
37		nnre 12265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
38	37	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
39	38	3ad2antl2 1205	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
40	39	renegcld 11666	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ)
41	19	ffvelcdmda 7078	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
42	41	3ad2antl1 1204	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
43	42	renegcld 11666	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
44		n0i 4286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑋 → ¬ 𝑋 = ∅)
45		rncoss 5961	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (abs ∘ 𝑓) ⊆ ran abs
46		absf 15426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ abs:ℂ⟶ℝ
47		frn 6711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran abs ⊆ ℝ
49	45, 48	sstri 3940	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (abs ∘ 𝑓) ⊆ ℝ
50		ltso 11315	. . . . . . . . . . . . . . . . . . . . 21 ⊢ < Or ℝ
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → < Or ℝ)
52	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → abs:ℂ⟶ℝ)
53		ax-resscn 11182	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℝ ⊆ ℂ
54	53	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ℝ ⊆ ℂ)
55	52, 54, 19	fcoss 46041	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → (abs ∘ 𝑓):𝑋⟶ℝ)
56		hoicvr.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ∈ Fin)
57	56	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑋 ∈ Fin)
58		rnffi 46008	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ 𝑓):𝑋⟶ℝ ∧ 𝑋 ∈ Fin) → ran (abs ∘ 𝑓) ∈ Fin)
59	55, 57, 58	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ran (abs ∘ 𝑓) ∈ Fin)
60	59	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ∈ Fin)
61	18	frnd 6712	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → ran 𝑓 ⊆ ℝ)
62	46	fdmi 6715	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ dom abs = ℂ
63	62	eqcomi 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ℂ = dom abs
64	53, 63	sseqtri 3979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ℝ ⊆ dom abs
65	61, 64	sstrdi 3943	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → ran 𝑓 ⊆ dom abs)
66		dmcosseq 5962	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ran 𝑓 ⊆ dom abs → dom (abs ∘ 𝑓) = dom 𝑓)
67	65, 66	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom (abs ∘ 𝑓) = dom 𝑓)
68	18	fdmd 6714	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom 𝑓 = 𝑋)
69	67, 68	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → dom (abs ∘ 𝑓) = 𝑋)
70	69	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) = 𝑋)
71		neqne 2963	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
72	71	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
73	70, 72	eqnetrd 3022	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) ≠ ∅)
74	73	neneqd 2960	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ dom (abs ∘ 𝑓) = ∅)
75		dm0rn0 5908	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (dom (abs ∘ 𝑓) = ∅ ↔ ran (abs ∘ 𝑓) = ∅)
76	74, 75	sylnib 331	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ ran (abs ∘ 𝑓) = ∅)
77	76	neqned 2962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
78	77	adantll 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
79	49	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ⊆ ℝ)
80		fisupcl 9441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( < Or ℝ ∧ (ran (abs ∘ 𝑓) ∈ Fin ∧ ran (abs ∘ 𝑓) ≠ ∅ ∧ ran (abs ∘ 𝑓) ⊆ ℝ)) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
81	51, 60, 78, 79, 80	syl13anc 1399	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
82	49, 81	sselid 3929	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
83	44, 82	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
84	83	3ad2antl1 1204	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
85	18	ffvelcdmda 7078	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
86	85	recnd 11262	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℂ)
87	86	abscld 15527	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
88	87	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
89	88	3ad2antl1 1204	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
90	85	renegcld 11666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
91	90	leabsd 15503	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘-(𝑓‘𝑖)))
92	86	absnegd 15540	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘-(𝑓‘𝑖)) = (abs‘(𝑓‘𝑖)))
93	91, 92	breqtrd 5131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
94	93	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
95	94	3ad2antl1 1204	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
96	49	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ⊆ ℝ)
97	44, 78	sylan2 605	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
98	97	3ad2antl1 1204	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
99		fimaxre2 12185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ran (abs ∘ 𝑓) ⊆ ℝ ∧ ran (abs ∘ 𝑓) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
100	49, 59, 99	sylancr 599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
101	100	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
102	101	3ad2antl1 1204	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
103		elmapfun 8868	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → Fun 𝑓)
104		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
105	68	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → 𝑋 = dom 𝑓)
106	105	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑋 = dom 𝑓)
107	104, 106	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom 𝑓)
108		fvco 6977	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
109	103, 107, 108	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
110		absfun 46181	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Fun abs
111		funco 6574	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun abs ∧ Fun 𝑓) → Fun (abs ∘ 𝑓))
112	110, 103, 111	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ (ℝ ↑_m 𝑋) → Fun (abs ∘ 𝑓))
113	86, 63	eleqtrdi 2870	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ dom abs)
114		dmfco 6975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
115	103, 107, 114	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
116	113, 115	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom (abs ∘ 𝑓))
117		fvelrn 7070	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun (abs ∘ 𝑓) ∧ 𝑖 ∈ dom (abs ∘ 𝑓)) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
118	112, 116, 117	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
119	109, 118	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑_m 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
120	119	adantll 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
121	120	3ad2antl1 1204	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
122	96, 98, 102, 121	suprubd 12202	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
123	43, 89, 84, 95, 122	letrd 11392	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
124		simpl3 1212	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
125	43, 84, 39, 123, 124	lelttrd 11393	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) < 𝑗)
126	42, 39, 125	ltnegcon1d 11819	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 < (𝑓‘𝑖))
127	40, 42, 126	ltled 11383	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ≤ (𝑓‘𝑖))
128	42	leabsd 15503	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
129	42, 89, 84, 128, 122	letrd 11392	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
130	42, 84, 39, 129, 124	lelttrd 11393	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) < 𝑗)
131	29, 32, 36, 127, 130	elicod 13449	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
132	22, 23, 24, 25, 131	syl31anc 1400	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
133	132	adantl3r 763	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
134		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
135		fconstmpt 5717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑋 × {⟨-𝑗, 𝑗⟩}) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
136		snex 5404	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨-𝑗, 𝑗⟩} ∈ V
137	136	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {⟨-𝑗, 𝑗⟩} ∈ V)
138	56, 137	xpexd 7751	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑋 × {⟨-𝑗, 𝑗⟩}) ∈ V)
139	135, 138	eqeltrrid 2865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
140	139	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
141		hoicvr.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
142	141	fvmpt2 6999	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ℕ ∧ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
143	134, 140, 142	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
144	143	fveq1d 6881	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
145	144	3adant3 1150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
146		eqidd 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
147		eqidd 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑥 = 𝑖) → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
148		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ 𝑋)
149		opex 5439	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
150	149	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑋 → ⟨-𝑗, 𝑗⟩ ∈ V)
151	146, 147, 148, 150	fvmptd 6995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
152	151	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
153	145, 152	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ⟨-𝑗, 𝑗⟩)
154	153	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1^st ‘((𝐼‘𝑗)‘𝑖)) = (1^st ‘⟨-𝑗, 𝑗⟩))
155		negex 11480	. . . . . . . . . . . . . . . 16 ⊢ -𝑗 ∈ V
156		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑗 ∈ V
157	155, 156	op1st 7995	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘⟨-𝑗, 𝑗⟩) = -𝑗
158	154, 157	eqtrdi 2811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1^st ‘((𝐼‘𝑗)‘𝑖)) = -𝑗)
159	153	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2^nd ‘((𝐼‘𝑗)‘𝑖)) = (2^nd ‘⟨-𝑗, 𝑗⟩))
160	155, 156	op2nd 7996	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
161	159, 160	eqtrdi 2811	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2^nd ‘((𝐼‘𝑗)‘𝑖)) = 𝑗)
162	158, 161	oveq12d 7432	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))) = (-𝑗[,)𝑗))
163	162	eqcomd 2766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
164	163	3adant1r 1196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1^st ‘((𝐼‘𝑗)‘𝑖))[,)(2^nd ‘((𝐼‘𝑗)‘𝑖))))
165	164	ad5ant135 1394	. . . . . . . . . 10 ⊢ ((((((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))))
166	133, 165	eleqtrd 2862	. . . . . . . . 9 ⊢ ((((((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))))
167	26	zred 12726	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
168	167, 37	opelxpd 5694	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
169	168	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
170	143, 169	fmpt3d 7110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
171	170	ad4ant14 765	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
172	171	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
173		simpr 490	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
174	172, 173	fvovco 46026	. . . . . . . . 9 ⊢ ((((((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))))
175	166, 174	eleqtrrd 2863	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
176	175	ralrimiva 3154	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
177		vex 3454	. . . . . . . 8 ⊢ 𝑓 ∈ V
178	177	elixp 8912	. . . . . . 7 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖)))
179	21, 176, 178	sylanbrc 595	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
180	82	archd 45995	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
181	179, 180	reximddv3 3179	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
182	181	an32s 665	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
183	182	eliund 4958	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑_m 𝑋)) → 𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
184	183	ssd 45915	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
185	17, 184	pm2.61dan 825	1 ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3899 ∅c0 4279 {csn 4584 ⟨cop 4590 ∪ ciun 4951 class class class wbr 5103 ↦ cmpt 5186 Or wor 5562 × cxp 5653 dom cdm 5655 ran crn 5656 ∘ ccom 5659 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 ↑_m cmap 8827 Xcixp 8905 Fincfn 8953 supcsup 9411 ℂcc 11123 ℝcr 11124 ℝ^*cxr 11267 < clt 11268 ≤ cle 11269 -cneg 11467 ℕcn 12258 [,)cico 13401 abscabs 15322

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9413 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-rp 13044 df-ico 13405 df-seq 14067 df-exp 14127 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324

This theorem is used by: hoicvrrex 47385

W3C validator