smflimlem4 - Mathbox for Glauco Siliprandi

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Theorem smflimlem4 45005

Description: Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smflimlem4.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimlem4.2	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimlem4.3	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimlem4.4	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimlem4.5	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem4.6	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
smflimlem4.7	⊢ (𝜑 → 𝐴 ∈ ℝ)
smflimlem4.8	⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
smflimlem4.9	⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
smflimlem4.10	⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
smflimlem4.11	⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)

**Assertion**
Ref	Expression
smflimlem4	⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴})

Distinct variable groups: 𝐴,𝑘,𝑚,𝑠 𝑥,𝐴,𝑘,𝑚 𝐶,𝑘,𝑚,𝑠 𝐶,𝑟,𝑘 𝐷,𝑘,𝑚,𝑛,𝑥 𝐷,𝑟,𝑥 𝑘,𝐹,𝑚,𝑛,𝑥 𝐹,𝑠 𝑚,𝐺 𝑘,𝐻,𝑚,𝑛 𝑘,𝐼,𝑚,𝑥 𝐼,𝑟 𝑚,𝑀 𝑃,𝑘,𝑚,𝑠 𝑃,𝑟 𝑆,𝑘,𝑚,𝑠 𝑘,𝑍,𝑚,𝑛,𝑥 𝜑,𝑘,𝑚,𝑛,𝑥 𝜑,𝑟 Allowed substitution hints: 𝜑(𝑠) 𝐴(𝑛,𝑟) 𝐶(𝑥,𝑛) 𝐷(𝑠) 𝑃(𝑥,𝑛) 𝑆(𝑥,𝑛,𝑟) 𝐹(𝑟) 𝐺(𝑥,𝑘,𝑛,𝑠,𝑟) 𝐻(𝑥,𝑠,𝑟) 𝐼(𝑛,𝑠) 𝑀(𝑥,𝑘,𝑛,𝑠,𝑟) 𝑍(𝑠,𝑟)

Proof of Theorem smflimlem4 Dummy variables 𝑖 𝑗 𝑧 𝑦 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4188	. . 3 ⊢ (𝐷 ∩ 𝐼) ⊆ 𝐷
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ 𝐷)
3	2	sselda 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → 𝑥 ∈ 𝐷)
4		smflimlem4.6	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
5	4	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))))
6		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑥 ∈ 𝐷)
7		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝐹
8		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑧𝐹
9		smflimlem4.2	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10		smflimlem4.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
12		smflimlem4.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
13	12	ffvelcdmda 7035	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
14		eqid 2736	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
15	11, 13, 14	smff 44963	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
16	15	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
17		smflimlem4.5	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
18		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
19	18	mpteq2dv 5207	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)))
20	19	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ ))
21	20	cbvrabv 3417	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑧 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ }
22	17, 21	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑧 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑧)) ∈ dom ⇝ }
23		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
24	6, 7, 8, 9, 16, 22, 23	fnlimfvre 43905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)
25	24	elexd 3465	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ V)
26	5, 25	fvmpt2d 6961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
27	26, 24	eqeltrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) ∈ ℝ)
28	3, 27	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ∈ ℝ)
29	28	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝐺‘𝑥) ∈ ℝ)
30		smflimlem4.7	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
31	30	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
32		rpre 12923	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ)
33	32	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
34	31, 33	readdcld 11184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝐴 + 𝑦) ∈ ℝ)
35	34	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝐴 + 𝑦) ∈ ℝ)
36		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺)
37		rphalfcl 12942	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 / 2) ∈ ℝ⁺)
38		rpgtrecnn 43604	. . . . . . . . . . 11 ⊢ ((𝑦 / 2) ∈ ℝ⁺ → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝑦 / 2))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ⁺ → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝑦 / 2))
40	39	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝑦 / 2))
41	10	ad4antr 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → 𝑆 ∈ SAlg)
42	13	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
43	42	ad5ant15 757	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
44	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → 𝐴 ∈ ℝ)
45	44	ad3antrrr 728	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → 𝐴 ∈ ℝ)
46		smflimlem4.8	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
47		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝑍
48		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑍
49		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
50		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘{𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))}
51	18	breq1d 5115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))))
52	51	cbvrabv 3417	. . . . . . . . . . . . . . . . 17 ⊢ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))}
53	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))})
54		oveq2 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → (1 / 𝑘) = (1 / 𝑗))
55	54	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑗 → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝑗)))
56	55	breq2d 5117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑗 → (((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))))
57	56	rabbidv 3415	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))})
58	53, 57	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))})
59	58	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)) ↔ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))))
60	59	rabbidv 3415	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))})
61	47, 48, 49, 50, 60	cbvmpo2 43297	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))})
62	46, 61	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑧 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑧) < (𝐴 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))})
63		smflimlem4.9	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
64		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐶‘(𝑚𝑃𝑘))
65		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐶‘(𝑚𝑃𝑗))
66		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑚𝑃𝑘) = (𝑚𝑃𝑗))
67	66	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑚𝑃𝑗)))
68	47, 48, 64, 65, 67	cbvmpo2 43297	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))) = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑗)))
69	63, 68	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑗)))
70		smflimlem4.10	. . . . . . . . . . . 12 ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
71		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑘 = 𝑗)
72	71	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝑚𝐻𝑘) = (𝑚𝐻𝑗))
73	72	iineq2dv 4979	. . . . . . . . . . . . . 14 ⊢ ((𝑘 = 𝑗 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑗))
74	73	iuneq2dv 4978	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑗))
75	74	cbviinv 5001	. . . . . . . . . . . 12 ⊢ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑗 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑗)
76	70, 75	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝐼 = ∩ 𝑗 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑗)
77		smflimlem4.11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)
78	77	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)
79	78	ad5ant15 757	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)
80		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → 𝑥 ∈ (𝐷 ∩ 𝐼))
81		simplr 767	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → 𝑘 ∈ ℕ)
82	37	ad3antlr 729	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → (𝑦 / 2) ∈ ℝ⁺)
83		simpr 485	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → (1 / 𝑘) < (𝑦 / 2))
84	9, 41, 43, 22, 45, 62, 69, 76, 79, 80, 81, 82, 83	smflimlem3 45004	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑘 ∈ ℕ) ∧ (1 / 𝑘) < (𝑦 / 2)) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2))))
85	84	rexlimdva2 3154	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑘 ∈ ℕ (1 / 𝑘) < (𝑦 / 2) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2)))))
86	40, 85	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2))))
87		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺)
88		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝐹
89		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
90		smflimlem4.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
91	90	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
92		eleq1w 2820	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
93	92	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
94		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝐹‘𝑚) = (𝐹‘𝑖))
95	94	dmeqd 5861	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → dom (𝐹‘𝑚) = dom (𝐹‘𝑖))
96	94, 95	feq12d 6656	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ))
97	93, 96	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑖 → (((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)))
98	97, 15	chvarvv 2002	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)
99	98	ad4ant14 750	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)
100		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑙 → (𝐹‘𝑚) = (𝐹‘𝑙))
101	100	dmeqd 5861	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 𝑙 → dom (𝐹‘𝑚) = dom (𝐹‘𝑙))
102	101	cbviinv 5001	. . . . . . . . . . . . . . 15 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙)
103	102	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙))
104	103	iuneq2i 4975	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙)
105		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑚))
106	105	iineq1d 43290	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑙 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑙))
107		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑖 → (𝐹‘𝑙) = (𝐹‘𝑖))
108	107	dmeqd 5861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑖 → dom (𝐹‘𝑙) = dom (𝐹‘𝑖))
109	108	cbviinv 5001	. . . . . . . . . . . . . . . 16 ⊢ ∩ 𝑙 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑙) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
110	109	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ∩ 𝑙 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑙) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖))
111	106, 110	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖))
112	111	cbviunv 5000	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑙) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
113	104, 112	eqtri 2764	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
114	113	rabeqi 3420	. . . . . . . . . . 11 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
115		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
116	115	fveq1d 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑚 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
117	116	cbvmptv 5218	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
118	117	eqcomi 2745	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥))
119	118	eleq1i 2828	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ )
120	119	rabbii 3413	. . . . . . . . . . 11 ⊢ {𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ }
121	17, 114, 120	3eqtri 2768	. . . . . . . . . 10 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) ∣ (𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)) ∈ dom ⇝ }
122	118	fveq2i 6845	. . . . . . . . . . . 12 ⊢ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥)))
123	122	mpteq2i 5210	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥))))
124	4, 123	eqtri 2764	. . . . . . . . . 10 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑖 ∈ 𝑍 ↦ ((𝐹‘𝑖)‘𝑥))))
125	3	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → 𝑥 ∈ 𝐷)
126	37	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 2) ∈ ℝ⁺)
127	87, 88, 89, 91, 9, 99, 121, 124, 125, 126	fnlimabslt 43910	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)))
128	29	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
129		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐹‘𝑖)‘𝑥) ∈ ℝ)
130	128, 129	resubcld 11583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℝ)
131	130	adantrr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℝ)
132	130	recnd 11183	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ∈ ℂ)
133	132	abscld 15321	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) ∈ ℝ)
134	133	adantrr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) ∈ ℝ)
135	32	rehalfcld 12400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 / 2) ∈ ℝ)
136	135	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝑦 / 2) ∈ ℝ)
137	131	leabsd 15299	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) ≤ (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))))
138	28	recnd 11183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ∈ ℂ)
139	138	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
140		recn 11141	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹‘𝑖)‘𝑥) ∈ ℝ → ((𝐹‘𝑖)‘𝑥) ∈ ℂ)
141	140	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → ((𝐹‘𝑖)‘𝑥) ∈ ℂ)
142	139, 141	abssubd 15338	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ ((𝐹‘𝑖)‘𝑥) ∈ ℝ) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) = (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))))
143	142	adantrr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) = (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))))
144		simprr 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))
145	143, 144	eqbrtrd 5127	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) < (𝑦 / 2))
146	145	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (abs‘((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥))) < (𝑦 / 2))
147	131, 134, 136, 137, 146	lelttrd 11313	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) < (𝑦 / 2))
148	29	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝐺‘𝑥) ∈ ℝ)
149		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → ((𝐹‘𝑖)‘𝑥) ∈ ℝ)
150	148, 149, 136	ltsubadd2d 11753	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (((𝐺‘𝑥) − ((𝐹‘𝑖)‘𝑥)) < (𝑦 / 2) ↔ (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))
151	147, 150	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ (((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2))) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))
152	151	ex 413	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ((((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))
153	152	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → (𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))
154	153	ralimdva 3164	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) → (∀𝑖 ∈ (ℤ_≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))
155	154	ex 413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝑚 ∈ 𝑍 → (∀𝑖 ∈ (ℤ_≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))))
156	36, 155	reximdai 3244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(((𝐹‘𝑖)‘𝑥) ∈ ℝ ∧ (abs‘(((𝐹‘𝑖)‘𝑥) − (𝐺‘𝑥))) < (𝑦 / 2)) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2))))
157	127, 156	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)))
158	115	dmeqd 5861	. . . . . . . . . 10 ⊢ (𝑖 = 𝑚 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚))
159	158	eleq2d 2823	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → (𝑥 ∈ dom (𝐹‘𝑖) ↔ 𝑥 ∈ dom (𝐹‘𝑚)))
160	116	breq1d 5115	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → (((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))
161	159, 160	anbi12d 631	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → ((𝑥 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))))
162	116	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑖 = 𝑚 → (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)) = (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)))
163	162	breq2d 5117	. . . . . . . 8 ⊢ (𝑖 = 𝑚 → ((𝐺‘𝑥) < (((𝐹‘𝑖)‘𝑥) + (𝑦 / 2)) ↔ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))))
164	36, 9, 86, 157, 161, 163	rexanuz3 43296	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑚 ∈ 𝑍 ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))))
165		df-3an 1089	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))))
166		3ancomb 1099	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))
167	165, 166	bitr3i 276	. . . . . . . 8 ⊢ (((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))
168	167	rexbii 3097	. . . . . . 7 ⊢ (∃𝑚 ∈ 𝑍 ((𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2))) ↔ ∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))
169	164, 168	sylib 217	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))))
170	29	ad2antrr 724	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) ∈ ℝ)
171	15	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
172		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑥 ∈ dom (𝐹‘𝑚))
173	171, 172	ffvelcdmd 7036	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
174	173	ad4ant134 1174	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
175		simpllr 774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑦 ∈ ℝ⁺)
176	175, 135	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝑦 / 2) ∈ ℝ)
177	174, 176	readdcld 11184	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ)
178	177	adantl3r 748	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ)
179	178	3ad2antr1 1188	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∈ ℝ)
180		rehalfcl 12379	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → (𝑦 / 2) ∈ ℝ)
181	33, 180	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 2) ∈ ℝ)
182	31, 181	jca 512	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝐴 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ))
183		readdcl 11134	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) → (𝐴 + (𝑦 / 2)) ∈ ℝ)
184	182, 183	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝐴 + (𝑦 / 2)) ∈ ℝ)
185	184, 181	readdcld 11184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) ∈ ℝ)
186	185	ad5ant13 755	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) ∈ ℝ)
187		simpr2 1195	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)))
188	174	adantrr 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
189	184	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐴 + (𝑦 / 2)) ∈ ℝ)
190	176	adantrr 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝑦 / 2) ∈ ℝ)
191		simprr 771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))
192	188, 189, 190, 191	ltadd1dd 11766	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)))
193	192	adantl3r 748	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)))
194	193	3adantr2 1170	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)))
195	170, 179, 186, 187, 194	lttrd 11316	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)))
196	31	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
197	181	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 / 2) ∈ ℂ)
198	196, 197, 197	addassd 11177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + ((𝑦 / 2) + (𝑦 / 2))))
199	32	recnd 11183	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℂ)
200		2halves 12381	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → ((𝑦 / 2) + (𝑦 / 2)) = 𝑦)
201	199, 200	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ⁺ → ((𝑦 / 2) + (𝑦 / 2)) = 𝑦)
202	201	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ → (𝐴 + ((𝑦 / 2) + (𝑦 / 2))) = (𝐴 + 𝑦))
203	202	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝐴 + ((𝑦 / 2) + (𝑦 / 2))) = (𝐴 + 𝑦))
204	198, 203	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + 𝑦))
205	204	ad5ant13 755	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → ((𝐴 + (𝑦 / 2)) + (𝑦 / 2)) = (𝐴 + 𝑦))
206	195, 205	breqtrd 5131	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑚 ∈ 𝑍) ∧ (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2)))) → (𝐺‘𝑥) < (𝐴 + 𝑦))
207	206	rexlimdva2 3154	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑚 ∈ 𝑍 (𝑥 ∈ dom (𝐹‘𝑚) ∧ (𝐺‘𝑥) < (((𝐹‘𝑚)‘𝑥) + (𝑦 / 2)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (𝑦 / 2))) → (𝐺‘𝑥) < (𝐴 + 𝑦)))
208	169, 207	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝐺‘𝑥) < (𝐴 + 𝑦))
209	29, 35, 208	ltled 11303	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) ∧ 𝑦 ∈ ℝ⁺) → (𝐺‘𝑥) ≤ (𝐴 + 𝑦))
210	209	ralrimiva 3143	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → ∀𝑦 ∈ ℝ⁺ (𝐺‘𝑥) ≤ (𝐴 + 𝑦))
211		alrple 13125	. . . 4 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐺‘𝑥) ≤ 𝐴 ↔ ∀𝑦 ∈ ℝ⁺ (𝐺‘𝑥) ≤ (𝐴 + 𝑦)))
212	28, 44, 211	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → ((𝐺‘𝑥) ≤ 𝐴 ↔ ∀𝑦 ∈ ℝ⁺ (𝐺‘𝑥) ≤ (𝐴 + 𝑦)))
213	210, 212	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐷 ∩ 𝐼)) → (𝐺‘𝑥) ≤ 𝐴)
214	2, 213	ssrabdv 4031	1 ⊢ (𝜑 → (𝐷 ∩ 𝐼) ⊆ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∪ ciun 4954 ∩ ciin 4955 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℂcc 11049 ℝcr 11050 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 abscabs 15119 ⇝ cli 15366 SAlgcsalg 44539 SMblFncsmblfn 44926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-ioo 13268 df-ico 13270 df-fl 13697 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-smblfn 44927

This theorem is referenced by: smflimlem5 45006

W3C validator