Theorem hoicvr 39438

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 9906	. . . . . . 7 ⊢ ℝ ∈ V
2		mapdm0 38378	. . . . . . 7 ⊢ (ℝ ∈ V → (ℝ ↑𝑚 ∅) = {∅})
3	1, 2	ax-mp 5	. . . . . 6 ⊢ (ℝ ↑𝑚 ∅) = {∅}
4	3	a1i 11	. . . . 5 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 ∅) = {∅})
5		oveq2 6557	. . . . 5 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = (ℝ ↑𝑚 ∅))
6		ixpeq1 7805	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
7	6	iuneq2d 4483	. . . . . 6 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖))
8		ixp0x 7822	. . . . . . . . . 10 ⊢ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
9	8	a1i 11	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ → X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
10	9	iuneq2i 4475	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = ∪ 𝑗 ∈ ℕ {∅}
11		1nn 10908	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
12	11	ne0ii 3882	. . . . . . . . 9 ⊢ ℕ ≠ ∅
13		iunconst 4465	. . . . . . . . 9 ⊢ (ℕ ≠ ∅ → ∪ 𝑗 ∈ ℕ {∅} = {∅})
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ {∅} = {∅}
15	10, 14	eqtri 2632	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅}
16	15	a1i 11	. . . . . 6 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ ∅ (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
17	7, 16	eqtrd 2644	. . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) = {∅})
18	4, 5, 17	3eqtr4d 2654	. . . 4 ⊢ (𝑋 = ∅ → (ℝ ↑𝑚 𝑋) = ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
19		eqimss 3620	. . . 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 786	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝜑)
23		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 ∈ (ℝ ↑𝑚 𝑋))
24		simplr 788	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → ¬ 𝑋 = ∅)
25		rncoss 5307	. . . . . . . . . . 11 ⊢ ran (abs ∘ 𝑓) ⊆ ran abs
26		absf 13925	. . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ
27		frn 5966	. . . . . . . . . . . 12 ⊢ (abs:ℂ⟶ℝ → ran abs ⊆ ℝ)
28	26, 27	ax-mp 5	. . . . . . . . . . 11 ⊢ ran abs ⊆ ℝ
29	25, 28	sstri 3577	. . . . . . . . . 10 ⊢ ran (abs ∘ 𝑓) ⊆ ℝ
30	29	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ⊆ ℝ)
31		ltso 9997	. . . . . . . . . . 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 7765	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → 𝑓:𝑋⟶ℝ)
35	34	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓:𝑋⟶ℝ)
36		ax-resscn 9872	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → ℝ ⊆ ℂ)
38	35, 37	fssd 5970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓:𝑋⟶ℂ)
39		fco 5971	. . . . . . . . . . . . 13 ⊢ ((abs:ℂ⟶ℝ ∧ 𝑓:𝑋⟶ℂ) → (abs ∘ 𝑓):𝑋⟶ℝ)
40	33, 38, 39	syl2anc 691	. . . . . . . . . . . 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 38351	. . . . . . . . . . . 12 ⊢ (((abs ∘ 𝑓):𝑋⟶ℝ ∧ 𝑋 ∈ Fin) → ran (abs ∘ 𝑓) ∈ Fin)
44	40, 42, 43	syl2anc 691	. . . . . . . . . . 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		frn 5966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝑋⟶ℝ → ran 𝑓 ⊆ ℝ)
47	34, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → ran 𝑓 ⊆ ℝ)
48	26	fdmi 5965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom abs = ℂ
49	48	eqcomi 2619	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = dom abs
50	36, 49	sseqtri 3600	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ dom abs
51	50	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → ℝ ⊆ dom abs)
52	47, 51	sstrd 3578	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → ran 𝑓 ⊆ dom abs)
53		dmcosseq 5308	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑓 ⊆ dom abs → dom (abs ∘ 𝑓) = dom 𝑓)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → dom (abs ∘ 𝑓) = dom 𝑓)
55		fdm 5964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝑋⟶ℝ → dom 𝑓 = 𝑋)
56	34, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → dom 𝑓 = 𝑋)
57	54, 56	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → dom (abs ∘ 𝑓) = 𝑋)
58	57	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) = 𝑋)
59		neqne 2790	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
60	59	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
61	58, 60	eqnetrd 2849	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → dom (abs ∘ 𝑓) ≠ ∅)
62	61	neneqd 2787	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ dom (abs ∘ 𝑓) = ∅)
63		dm0rn0 5263	. . . . . . . . . . . . 13 ⊢ (dom (abs ∘ 𝑓) = ∅ ↔ ran (abs ∘ 𝑓) = ∅)
64	62, 63	sylnib 317	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → ¬ ran (abs ∘ 𝑓) = ∅)
65	64	neqned 2789	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
66	65	adantll 746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → ran (abs ∘ 𝑓) ≠ ∅)
67		fisupcl 8258	. . . . . . . . . 10 ⊢ (( < Or ℝ ∧ (ran (abs ∘ 𝑓) ∈ Fin ∧ ran (abs ∘ 𝑓) ≠ ∅ ∧ ran (abs ∘ 𝑓) ⊆ ℝ)) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
68	32, 45, 66, 30, 67	syl13anc 1320	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ran (abs ∘ 𝑓))
69	30, 68	sseldd 3569	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
70		arch 11166	. . . . . . . 8 ⊢ (sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ → ∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
71	69, 70	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
72	35	ffnd 5959	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 Fn 𝑋)
73	72	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 Fn 𝑋)
74	73	adantlr 747	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 Fn 𝑋)
75		simplll 794	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)))
76		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
77	76	ad3antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℕ)
78		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
79		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
80		simp2 1055	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑗 ∈ ℕ)
81		zssre 11261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℤ ⊆ ℝ
82		ressxr 9962	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℝ*
83	81, 82	sstri 3577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℤ ⊆ ℝ*
84		nnnegz 11257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℤ)
85	83, 84	sseldi 3566	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ*)
86	85	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ*)
87	80, 86	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ*)
88	76	nnxrd 38224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ*)
89	88	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ*)
90	80, 89	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ*)
91	34	3ad2ant1 1075	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ)
92	82	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → ℝ ⊆ ℝ*)
93	91, 92	fssd 5970	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ*)
94	93	3adant1l 1310	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓:𝑋⟶ℝ*)
95	94	ffvelrnda 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ*)
96		nnre 10904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℝ)
97	96	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
98	97	3ad2antl2 1217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑗 ∈ ℝ)
99	98	renegcld 10336	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ∈ ℝ)
100	35	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
101	100	3ad2antl1 1216	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
102	101	renegcld 10336	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
103		simpll 786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝜑)
104		simplr 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑𝑚 𝑋))
105		n0i 3879	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ 𝑋 → ¬ 𝑋 = ∅)
106	105	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → ¬ 𝑋 = ∅)
107	103, 104, 106, 69	syl21anc 1317	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
108	107	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) ∈ ℝ)
109	34	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℝ)
110	36, 109	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℂ)
111	110	abscld 14023	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
112	111	adantll 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
113	112	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ℝ)
114	109	renegcld 10336	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ∈ ℝ)
115	114	leabsd 14001	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘-(𝑓‘𝑖)))
116	110	absnegd 14036	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘-(𝑓‘𝑖)) = (abs‘(𝑓‘𝑖)))
117	115, 116	breqtrd 4609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
118	117	adantll 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
119	118	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
120	29	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ⊆ ℝ)
121	106, 66	syldan 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
122	121	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ran (abs ∘ 𝑓) ≠ ∅)
123		fimaxre2 10848	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ran (abs ∘ 𝑓) ⊆ ℝ ∧ ran (abs ∘ 𝑓) ∈ Fin) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
124	29, 44, 123	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
125	124	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
126	125	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (abs ∘ 𝑓)𝑧 ≤ 𝑦)
127		elmapfun 7767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → Fun 𝑓)
128	127	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → Fun 𝑓)
129		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
130	56	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → 𝑋 = dom 𝑓)
131	130	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑋 = dom 𝑓)
132	129, 131	eleqtrd 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom 𝑓)
133		fvco 6184	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
134	128, 132, 133	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) = (abs‘(𝑓‘𝑖)))
135	134	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) = ((abs ∘ 𝑓)‘𝑖))
136		absfun 38507	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ Fun abs
137	136	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → Fun abs)
138		funco 5842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun abs ∧ Fun 𝑓) → Fun (abs ∘ 𝑓))
139	137, 127, 138	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 ∈ (ℝ ↑𝑚 𝑋) → Fun (abs ∘ 𝑓))
140	139	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → Fun (abs ∘ 𝑓))
141	110, 49	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ dom abs)
142		dmfco 6182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun 𝑓 ∧ 𝑖 ∈ dom 𝑓) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
143	128, 132, 142	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (𝑖 ∈ dom (abs ∘ 𝑓) ↔ (𝑓‘𝑖) ∈ dom abs))
144	141, 143	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ dom (abs ∘ 𝑓))
145		fvelrn 6260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun (abs ∘ 𝑓) ∧ 𝑖 ∈ dom (abs ∘ 𝑓)) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
146	140, 144, 145	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → ((abs ∘ 𝑓)‘𝑖) ∈ ran (abs ∘ 𝑓))
147	135, 146	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ∈ (ℝ ↑𝑚 𝑋) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
148	147	adantll 746	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
149	148	3ad2antl1 1216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ∈ ran (abs ∘ 𝑓))
150		suprub 10863	. . . . . . . . . . . . . . . . . . . . . . 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, < ))
151	120, 122, 126, 149, 150	syl31anc 1321	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (abs‘(𝑓‘𝑖)) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
152	102, 113, 108, 119, 151	letrd 10073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
153		simpl3 1059	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗)
154	102, 108, 98, 152, 153	lelttrd 10074	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -(𝑓‘𝑖) < 𝑗)
155	102, 98	ltnegd 10484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (-(𝑓‘𝑖) < 𝑗 ↔ -𝑗 < --(𝑓‘𝑖)))
156	154, 155	mpbid 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 < --(𝑓‘𝑖))
157	36, 101	sseldi 3566	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ℂ)
158	157	negnegd 10262	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → --(𝑓‘𝑖) = (𝑓‘𝑖))
159	156, 158	breqtrd 4609	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 < (𝑓‘𝑖))
160	99, 101, 159	ltled 10064	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → -𝑗 ≤ (𝑓‘𝑖))
161	101	leabsd 14001	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ (abs‘(𝑓‘𝑖)))
162	101, 113, 108, 161, 151	letrd 10073	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ≤ sup(ran (abs ∘ 𝑓), ℝ, < ))
163	101, 108, 98, 162, 153	lelttrd 10074	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) < 𝑗)
164	87, 90, 95, 160, 163	elicod 12095	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
165	75, 77, 78, 79, 164	syl31anc 1321	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
166	165	adantlllr 38222	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (-𝑗[,)𝑗))
167		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
168		mptexg 6389	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ Fin → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
169	41, 168	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
170	169	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V)
171		hoicvr.2	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
172	171	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ ∧ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) ∈ V) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
173	167, 170, 172	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
174	173	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
175	174	3adant3 1074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖))
176		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩))
177		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩
178	177	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ 𝑋 ∧ 𝑥 = 𝑖) → ⟨-𝑗, 𝑗⟩ = ⟨-𝑗, 𝑗⟩)
179		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ 𝑋)
180		opex 4859	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⟨-𝑗, 𝑗⟩ ∈ V
181	180	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝑋 → ⟨-𝑗, 𝑗⟩ ∈ V)
182	176, 178, 179, 181	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ 𝑋 → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
183	182	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)‘𝑖) = ⟨-𝑗, 𝑗⟩)
184	175, 183	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑖) = ⟨-𝑗, 𝑗⟩)
185	184	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑖)) = (1st ‘⟨-𝑗, 𝑗⟩))
186		negex 10158	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝑗 ∈ V
187		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑗 ∈ V
188	186, 187	op1st 7067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗
189	188	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1st ‘⟨-𝑗, 𝑗⟩) = -𝑗)
190	185, 189	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑖)) = -𝑗)
191	184	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑖)) = (2nd ‘⟨-𝑗, 𝑗⟩))
192	186, 187	op2nd 7068	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗
193	192	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2nd ‘⟨-𝑗, 𝑗⟩) = 𝑗)
194	191, 193	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑖)) = 𝑗)
195	190, 194	oveq12d 6567	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))) = (-𝑗[,)𝑗))
196	195	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))))
197	196	3adant1r 1311	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ 𝑗 ∈ ℕ ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))))
198	197	ad5ant135 1306	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (-𝑗[,)𝑗) = ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))))
199	166, 198	eleqtrd 2690	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))))
200	81, 84	sseldi 3566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ → -𝑗 ∈ ℝ)
201		opelxpi 5072	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-𝑗 ∈ ℝ ∧ 𝑗 ∈ ℝ) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
202	200, 96, 201	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ ℕ → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
203	202	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ 𝑋) → ⟨-𝑗, 𝑗⟩ ∈ (ℝ × ℝ))
204		eqid 2610	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩) = (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)
205	203, 204	fmptd 6292	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ))
206	173	feq1d 5943	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐼‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩):𝑋⟶(ℝ × ℝ)))
207	205, 206	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
208	207	ad4ant14 1285	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
209	208	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
210		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
211	209, 210	fvovco 38376	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑖) = ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))))
212	211	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑖))[,)(2nd ‘((𝐼‘𝑗)‘𝑖))) = (([,) ∘ (𝐼‘𝑗))‘𝑖))
213	199, 212	eleqtrd 2690	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) ∧ 𝑖 ∈ 𝑋) → (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
214	213	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖))
215	74, 214	jca 553	. . . . . . . . . 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)))
216		vex 3176	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
217	216	elixp 7801	. . . . . . . . . 10 ⊢ (𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝑓‘𝑖) ∈ (([,) ∘ (𝐼‘𝑗))‘𝑖)))
218	215, 217	sylibr 223	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) ∧ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗) → 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
219	218	ex 449	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ ℕ) → (sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗 → 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)))
220	219	reximdva 3000	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → (∃𝑗 ∈ ℕ sup(ran (abs ∘ 𝑓), ℝ, < ) < 𝑗 → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖)))
221	71, 220	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) ∧ ¬ 𝑋 = ∅) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
222	22, 23, 24, 221	syl21anc 1317	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
223		eliun 4460	. . . . 5 ⊢ (𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
224	222, 223	sylibr 223	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑓 ∈ (ℝ ↑𝑚 𝑋)) → 𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
225	224	ralrimiva 2949	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∀𝑓 ∈ (ℝ ↑𝑚 𝑋)𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
226		dfss3 3558	. . 3 ⊢ ((ℝ ↑𝑚 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖) ↔ ∀𝑓 ∈ (ℝ ↑𝑚 𝑋)𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
227	225, 226	sylibr 223	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (ℝ ↑𝑚 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))
228	21, 227	pm2.61dan 828	1 ⊢ (𝜑 → (ℝ ↑𝑚 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   Or wor 4958   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  supcsup 8229  ℂcc 9813  ℝcr 9814  1c1 9816  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  -cneg 10146  ℕcn 10897  ℤcz 11254  [,)cico 12048  abscabs 13822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824

This theorem is referenced by:  hoicvrrex  39446