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Theorem nntopi 7174
 Description: Mapping from ℕ to N. (Contributed by Jim Kingdon, 13-Jul-2021.)
Hypothesis
Ref Expression
nntopi.n 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
Assertion
Ref Expression
nntopi (𝐴𝑁 → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑧,𝐴   𝑧,𝑁,𝑦,𝑥   𝑢,𝑙,𝑧,𝑦,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑢,𝑙)   𝑁(𝑢,𝑙)

Proof of Theorem nntopi
Dummy variables 𝑤 𝑘 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nntopi.n . 2 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
2 eqeq2 2092 . . 3 (𝑤 = 1 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1))
32rexbidv 2374 . 2 (𝑤 = 1 → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1))
4 eqeq2 2092 . . 3 (𝑤 = 𝑘 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘))
54rexbidv 2374 . 2 (𝑤 = 𝑘 → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘))
6 eqeq2 2092 . . 3 (𝑤 = (𝑘 + 1) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
76rexbidv 2374 . 2 (𝑤 = (𝑘 + 1) → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
8 eqeq2 2092 . . 3 (𝑤 = 𝐴 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴))
98rexbidv 2374 . 2 (𝑤 = 𝐴 → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴))
10 1pi 6619 . . 3 1𝑜N
11 eqid 2083 . . 3 1 = 1
12 opeq1 3590 . . . . . . . . . . . . . . . . 17 (𝑧 = 1𝑜 → ⟨𝑧, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
1312eceq1d 6229 . . . . . . . . . . . . . . . 16 (𝑧 = 1𝑜 → [⟨𝑧, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
14 df-1nqqs 6655 . . . . . . . . . . . . . . . 16 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
1513, 14syl6eqr 2133 . . . . . . . . . . . . . . 15 (𝑧 = 1𝑜 → [⟨𝑧, 1𝑜⟩] ~Q = 1Q)
1615breq2d 3817 . . . . . . . . . . . . . 14 (𝑧 = 1𝑜 → (𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q𝑙 <Q 1Q))
1716abbidv 2200 . . . . . . . . . . . . 13 (𝑧 = 1𝑜 → {𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q 1Q})
1815breq1d 3815 . . . . . . . . . . . . . 14 (𝑧 = 1𝑜 → ([⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
1918abbidv 2200 . . . . . . . . . . . . 13 (𝑧 = 1𝑜 → {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
2017, 19opeq12d 3598 . . . . . . . . . . . 12 (𝑧 = 1𝑜 → ⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩)
21 df-i1p 6771 . . . . . . . . . . . 12 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
2220, 21syl6eqr 2133 . . . . . . . . . . 11 (𝑧 = 1𝑜 → ⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ = 1P)
2322oveq1d 5578 . . . . . . . . . 10 (𝑧 = 1𝑜 → (⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
2423opeq1d 3596 . . . . . . . . 9 (𝑧 = 1𝑜 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩)
2524eceq1d 6229 . . . . . . . 8 (𝑧 = 1𝑜 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
26 df-1r 7023 . . . . . . . 8 1R = [⟨(1P +P 1P), 1P⟩] ~R
2725, 26syl6eqr 2133 . . . . . . 7 (𝑧 = 1𝑜 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
2827opeq1d 3596 . . . . . 6 (𝑧 = 1𝑜 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨1R, 0R⟩)
29 df-1 7103 . . . . . 6 1 = ⟨1R, 0R
3028, 29syl6eqr 2133 . . . . 5 (𝑧 = 1𝑜 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
3130eqeq1d 2091 . . . 4 (𝑧 = 1𝑜 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1 ↔ 1 = 1))
3231rspcev 2710 . . 3 ((1𝑜N ∧ 1 = 1) → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
3310, 11, 32mp2an 417 . 2 𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
34 simplr 497 . . . . . . 7 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → 𝑧N)
35 addclpi 6631 . . . . . . 7 ((𝑧N ∧ 1𝑜N) → (𝑧 +N 1𝑜) ∈ N)
3634, 10, 35sylancl 404 . . . . . 6 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (𝑧 +N 1𝑜) ∈ N)
37 pitonnlem2 7129 . . . . . . . 8 (𝑧N → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
3834, 37syl 14 . . . . . . 7 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
39 simpr 108 . . . . . . . 8 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘)
4039oveq1d 5578 . . . . . . 7 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = (𝑘 + 1))
4138, 40eqtr3d 2117 . . . . . 6 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
42 opeq1 3590 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧 +N 1𝑜) → ⟨𝑣, 1𝑜⟩ = ⟨(𝑧 +N 1𝑜), 1𝑜⟩)
4342eceq1d 6229 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧 +N 1𝑜) → [⟨𝑣, 1𝑜⟩] ~Q = [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q )
4443breq2d 3817 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1𝑜) → (𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q ))
4544abbidv 2200 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1𝑜) → {𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q })
4643breq1d 3815 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1𝑜) → ([⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢))
4746abbidv 2200 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1𝑜) → {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢})
4845, 47opeq12d 3598 . . . . . . . . . . . 12 (𝑣 = (𝑧 +N 1𝑜) → ⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩)
4948oveq1d 5578 . . . . . . . . . . 11 (𝑣 = (𝑧 +N 1𝑜) → (⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
5049opeq1d 3596 . . . . . . . . . 10 (𝑣 = (𝑧 +N 1𝑜) → ⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
5150eceq1d 6229 . . . . . . . . 9 (𝑣 = (𝑧 +N 1𝑜) → [⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
5251opeq1d 3596 . . . . . . . 8 (𝑣 = (𝑧 +N 1𝑜) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
5352eqeq1d 2091 . . . . . . 7 (𝑣 = (𝑧 +N 1𝑜) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
5453rspcev 2710 . . . . . 6 (((𝑧 +N 1𝑜) ∈ N ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)) → ∃𝑣N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
5536, 41, 54syl2anc 403 . . . . 5 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ∃𝑣N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
5655ex 113 . . . 4 ((𝑘𝑁𝑧N) → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑣N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
5756rexlimdva 2482 . . 3 (𝑘𝑁 → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑣N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
58 opeq1 3590 . . . . . . . . . . . . 13 (𝑣 = 𝑧 → ⟨𝑣, 1𝑜⟩ = ⟨𝑧, 1𝑜⟩)
5958eceq1d 6229 . . . . . . . . . . . 12 (𝑣 = 𝑧 → [⟨𝑣, 1𝑜⟩] ~Q = [⟨𝑧, 1𝑜⟩] ~Q )
6059breq2d 3817 . . . . . . . . . . 11 (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q ))
6160abbidv 2200 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q })
6259breq1d 3815 . . . . . . . . . . 11 (𝑣 = 𝑧 → ([⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢))
6362abbidv 2200 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢})
6461, 63opeq12d 3598 . . . . . . . . 9 (𝑣 = 𝑧 → ⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩)
6564oveq1d 5578 . . . . . . . 8 (𝑣 = 𝑧 → (⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
6665opeq1d 3596 . . . . . . 7 (𝑣 = 𝑧 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
6766eceq1d 6229 . . . . . 6 (𝑣 = 𝑧 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
6867opeq1d 3596 . . . . 5 (𝑣 = 𝑧 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
6968eqeq1d 2091 . . . 4 (𝑣 = 𝑧 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
7069cbvrexv 2583 . . 3 (∃𝑣N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
7157, 70syl6ib 159 . 2 (𝑘𝑁 → (∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
721, 3, 5, 7, 9, 33, 71nnindnn 7173 1 (𝐴𝑁 → ∃𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   = wceq 1285   ∈ wcel 1434  {cab 2069  ∀wral 2353  ∃wrex 2354  ⟨cop 3419  ∩ cint 3656   class class class wbr 3805  (class class class)co 5563  1𝑜c1o 6078  [cec 6191  Ncnpi 6576   +N cpli 6577   ~Q ceq 6583  1Qc1q 6585
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