ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nntopi GIF version

Theorem nntopi 7895
Description: Mapping from 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 2187 . . 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 2478 . 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 2187 . . 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 2478 . 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 2187 . . 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 2478 . 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 2187 . . 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 2478 . 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 7316 . . 3 1oN
11 eqid 2177 . . 3 1 = 1
12 opeq1 3780 . . . . . . . . . . . . . . . . 17 (𝑧 = 1o → ⟨𝑧, 1o⟩ = ⟨1o, 1o⟩)
1312eceq1d 6573 . . . . . . . . . . . . . . . 16 (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
14 df-1nqqs 7352 . . . . . . . . . . . . . . . 16 1Q = [⟨1o, 1o⟩] ~Q
1513, 14eqtr4di 2228 . . . . . . . . . . . . . . 15 (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = 1Q)
1615breq2d 4017 . . . . . . . . . . . . . 14 (𝑧 = 1o → (𝑙 <Q [⟨𝑧, 1o⟩] ~Q𝑙 <Q 1Q))
1716abbidv 2295 . . . . . . . . . . . . 13 (𝑧 = 1o → {𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q } = {𝑙𝑙 <Q 1Q})
1815breq1d 4015 . . . . . . . . . . . . . 14 (𝑧 = 1o → ([⟨𝑧, 1o⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
1918abbidv 2295 . . . . . . . . . . . . 13 (𝑧 = 1o → {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
2017, 19opeq12d 3788 . . . . . . . . . . . 12 (𝑧 = 1o → ⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩)
21 df-i1p 7468 . . . . . . . . . . . 12 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
2220, 21eqtr4di 2228 . . . . . . . . . . 11 (𝑧 = 1o → ⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = 1P)
2322oveq1d 5892 . . . . . . . . . 10 (𝑧 = 1o → (⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
2423opeq1d 3786 . . . . . . . . 9 (𝑧 = 1o → ⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩)
2524eceq1d 6573 . . . . . . . 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 7733 . . . . . . . 8 1R = [⟨(1P +P 1P), 1P⟩] ~R
2725, 26eqtr4di 2228 . . . . . . 7 (𝑧 = 1o → [⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
2827opeq1d 3786 . . . . . 6 (𝑧 = 1o → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨1R, 0R⟩)
29 df-1 7821 . . . . . 6 1 = ⟨1R, 0R
3028, 29eqtr4di 2228 . . . . 5 (𝑧 = 1o → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
3130eqeq1d 2186 . . . 4 (𝑧 = 1o → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1 ↔ 1 = 1))
3231rspcev 2843 . . 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 426 . 2 𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
34 simplr 528 . . . . . . 7 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → 𝑧N)
35 addclpi 7328 . . . . . . 7 ((𝑧N ∧ 1oN) → (𝑧 +N 1o) ∈ N)
3634, 10, 35sylancl 413 . . . . . 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 7848 . . . . . . . 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 110 . . . . . . . 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 5892 . . . . . . 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 2212 . . . . . 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 3780 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧 +N 1o) → ⟨𝑣, 1o⟩ = ⟨(𝑧 +N 1o), 1o⟩)
4342eceq1d 6573 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧 +N 1o) → [⟨𝑣, 1o⟩] ~Q = [⟨(𝑧 +N 1o), 1o⟩] ~Q )
4443breq2d 4017 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1o) → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q ))
4544abbidv 2295 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1o) → {𝑙𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q })
4643breq1d 4015 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1o) → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢))
4746abbidv 2295 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1o) → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢})
4845, 47opeq12d 3788 . . . . . . . . . . . 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 5892 . . . . . . . . . . 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 3786 . . . . . . . . . 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 6573 . . . . . . . . 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 3786 . . . . . . . 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 2186 . . . . . . 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 2843 . . . . . 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 411 . . . . 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 115 . . . 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 2594 . . 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 3780 . . . . . . . . . . . . 13 (𝑣 = 𝑧 → ⟨𝑣, 1o⟩ = ⟨𝑧, 1o⟩)
5958eceq1d 6573 . . . . . . . . . . . 12 (𝑣 = 𝑧 → [⟨𝑣, 1o⟩] ~Q = [⟨𝑧, 1o⟩] ~Q )
6059breq2d 4017 . . . . . . . . . . 11 (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q𝑙 <Q [⟨𝑧, 1o⟩] ~Q ))
6160abbidv 2295 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑙𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑧, 1o⟩] ~Q })
6259breq1d 4015 . . . . . . . . . . 11 (𝑣 = 𝑧 → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1o⟩] ~Q <Q 𝑢))
6362abbidv 2295 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢})
6461, 63opeq12d 3788 . . . . . . . . 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 5892 . . . . . . . 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 3786 . . . . . . 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 6573 . . . . . 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 3786 . . . . 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 2186 . . . 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 2706 . . 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, 70imbitrdi 161 . 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 7894 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 104   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  cop 3597   cint 3846   class class class wbr 4005  (class class class)co 5877  1oc1o 6412  [cec 6535  Ncnpi 7273   +N cpli 7274   ~Q ceq 7280  1Qc1q 7282   <Q cltq 7286  1Pc1p 7293   +P cpp 7294   ~R cer 7297  0Rc0r 7299  1Rc1r 7300  1c1 7814   + caddc 7816
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-enr 7727  df-nr 7728  df-plr 7729  df-0r 7732  df-1r 7733  df-c 7819  df-1 7821  df-r 7823  df-add 7824
This theorem is referenced by:  axcaucvglemres  7900
  Copyright terms: Public domain W3C validator