Theorem recriota 7950

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 7910	. . 3 ⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ℝ)
2		pitoregt0 7909	. . 3 ⊢ (𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3		axprecex 7940	. . 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 411	. 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 540	. . . 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 529	. . . . 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 276	. . . . . 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 276	. . . . . 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 7949	. . . . . 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 411	. . . . 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 5926	. . . . . . 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 2202	. . . . . 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 5896	. . . . 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 411	. . . 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 147	. . 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 5934	. . . 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 7920	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
18	17, 7	sselid 3177	. . . . . . . . 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 7911	. . . . . . . . . . 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 276	. . . . . . . . . 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	sselid 3177	. . . . . . . . 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 7931	. . . . . . . . 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 411	. . . . . . . 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 7918	. . . . . . . . 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 276	. . . . . . . 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 2228	. . . . . . 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 5933	. . . . . 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	sselid 3177	. . . . . . 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 7933	. . . . . . 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 1249	. . . . . 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 7921	. . . . . . 7 ⊢ 1 ∈ ℂ
32		axmulcom 7931	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 · 𝑦) = (𝑦 · 1))
33	31, 28, 32	sylancr 414	. . . . . 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 2234	. . . . 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 7937	. . . . . 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 2226	. . . 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 7937	. . . . 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 2234	. . 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 2226	. 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 2616	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 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  ∃!wreu 2474  ⟨cop 3621   class class class wbr 4029  ‘cfv 5254  ℩crio 5872  (class class class)co 5918  1_oc1o 6462  [cec 6585  Ncnpi 7332   ~_Q ceq 7339  *_Qcrq 7344   <_Q cltq 7345  1_Pc1p 7352   +_P cpp 7353   ~_R cer 7356  0_Rc0r 7358  ℂcc 7870  ℝcr 7871  0cc0 7872  1c1 7873   <_ℝ cltrr 7876   · cmul 7877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-imp 7529  df-iltp 7530  df-enr 7786  df-nr 7787  df-plr 7788  df-mr 7789  df-ltr 7790  df-0r 7791  df-1r 7792  df-m1r 7793  df-c 7878  df-0 7879  df-1 7880  df-r 7882  df-mul 7884  df-lt 7885

This theorem is referenced by:  axcaucvglemcau  7958