Theorem recriota 7056

Description: Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)

Assertion

Ref	Expression
recriota	⊢ (𝑁 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Distinct variable group:   𝑁,𝑙,𝑟,𝑢

Proof of Theorem recriota

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pitore 7018	. . 3 ⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
2		pitoregt0 7017	. . 3 ⊢ (𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3		axprecex 7046	. . 3 ⊢ ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ ∧ 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → ∃𝑦 ∈ ℝ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))
4	1, 2, 3	syl2anc 403	. 2 ⊢ (𝑁 ∈ N → ∃𝑦 ∈ ℝ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))
5		simprrr 506	. . . 4 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1)
6		simprl 497	. . . . 5 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → 𝑦 ∈ ℝ)
7	1	adantr 270	. . . . . 6 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
8	2	adantr 270	. . . . . 6 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
9		rereceu 7055	. . . . . 6 ⊢ ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ ∧ 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → ∃!𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)
10	7, 8, 9	syl2anc 403	. . . . 5 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ∃!𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1)
11		oveq2 5540	. . . . . . 7 ⊢ (𝑟 = 𝑦 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦))
12	11	eqeq1d 2089	. . . . . 6 ⊢ (𝑟 = 𝑦 → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1 ↔ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))
13	12	riota2 5510	. . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ ∃!𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1 ↔ (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = 𝑦))
14	6, 10, 13	syl2anc 403	. . . 4 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1 ↔ (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = 𝑦))
15	5, 14	mpbid 145	. . 3 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = 𝑦)
16	5	oveq2d 5548	. . . 4 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦)) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 1))
17		axresscn 7028	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
18	17, 7	sseldi 2997	. . . . . . . . 9 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ)
19		recnnre 7019	. . . . . . . . . . 11 ⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
20	19	adantr 270	. . . . . . . . . 10 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
21	17, 20	sseldi 2997	. . . . . . . . 9 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ)
22		axmulcom 7037	. . . . . . . . 9 ⊢ ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
23	18, 21, 22	syl2anc 403	. . . . . . . 8 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
24		recidpirq 7026	. . . . . . . . 9 ⊢ (𝑁 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)
25	24	adantr 270	. . . . . . . 8 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)
26	23, 25	eqtr3d 2115	. . . . . . 7 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = 1)
27	26	oveq1d 5547	. . . . . 6 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) · 𝑦) = (1 · 𝑦))
28	17, 6	sseldi 2997	. . . . . . 7 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → 𝑦 ∈ ℂ)
29		axmulass 7039	. . . . . . 7 ⊢ ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) · 𝑦) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦)))
30	21, 18, 28, 29	syl3anc 1169	. . . . . 6 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) · 𝑦) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦)))
31		ax1cn 7029	. . . . . . 7 ⊢ 1 ∈ ℂ
32		axmulcom 7037	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 · 𝑦) = (𝑦 · 1))
33	31, 28, 32	sylancr 405	. . . . . 6 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (1 · 𝑦) = (𝑦 · 1))
34	27, 30, 33	3eqtr3d 2121	. . . . 5 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦)) = (𝑦 · 1))
35		ax1rid 7043	. . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝑦 · 1) = 𝑦)
36	6, 35	syl 14	. . . . 5 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (𝑦 · 1) = 𝑦)
37	34, 36	eqtrd 2113	. . . 4 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦)) = 𝑦)
38		ax1rid 7043	. . . . 5 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
39	20, 38	syl 14	. . . 4 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
40	16, 37, 39	3eqtr3d 2121	. . 3 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → 𝑦 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
41	15, 40	eqtrd 2113	. 2 ⊢ ((𝑁 ∈ N ∧ (𝑦 ∈ ℝ ∧ (0 <ℝ 𝑦 ∧ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑦) = 1))) → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
42	4, 41	rexlimddv 2481	1 ⊢ (𝑁 ∈ N → (℩𝑟 ∈ ℝ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ · 𝑟) = 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  {cab 2067  ∃wrex 2349  ∃!wreu 2350  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  ℩crio 5487  (class class class)co 5532  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   ~_Q ceq 6469  *_Qcrq 6474   <_Q cltq 6475  1_Pc1p 6482   +_P cpp 6483   ~_R cer 6486  0_Rc0r 6488  ℂcc 6979  ℝcr 6980  0cc0 6981  1c1 6982   <_ℝ cltrr 6985   · cmul 6986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-imp 6659  df-iltp 6660  df-enr 6903  df-nr 6904  df-plr 6905  df-mr 6906  df-ltr 6907  df-0r 6908  df-1r 6909  df-m1r 6910  df-c 6987  df-0 6988  df-1 6989  df-r 6991  df-mul 6993  df-lt 6994

This theorem is referenced by:  axcaucvglemcau  7064