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Theorem nntopi 7025
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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 6470 . . 3 1𝑜N
11 eqid 2056 . . 3 1 = 1
12 opeq1 3576 . . . . . . . . . . . . . . . . 17 (𝑧 = 1𝑜 → ⟨𝑧, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
1312eceq1d 6172 . . . . . . . . . . . . . . . 16 (𝑧 = 1𝑜 → [⟨𝑧, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
14 df-1nqqs 6506 . . . . . . . . . . . . . . . 16 1Q = [⟨1𝑜, 1𝑜⟩] ~Q
1513, 14syl6eqr 2106 . . . . . . . . . . . . . . 15 (𝑧 = 1𝑜 → [⟨𝑧, 1𝑜⟩] ~Q = 1Q)
1615breq2d 3803 . . . . . . . . . . . . . 14 (𝑧 = 1𝑜 → (𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q𝑙 <Q 1Q))
1716abbidv 2171 . . . . . . . . . . . . 13 (𝑧 = 1𝑜 → {𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q 1Q})
1815breq1d 3801 . . . . . . . . . . . . . 14 (𝑧 = 1𝑜 → ([⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
1918abbidv 2171 . . . . . . . . . . . . 13 (𝑧 = 1𝑜 → {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
2017, 19opeq12d 3584 . . . . . . . . . . . 12 (𝑧 = 1𝑜 → ⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩)
21 df-i1p 6622 . . . . . . . . . . . 12 1P = ⟨{𝑙𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
2220, 21syl6eqr 2106 . . . . . . . . . . 11 (𝑧 = 1𝑜 → ⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ = 1P)
2322oveq1d 5554 . . . . . . . . . 10 (𝑧 = 1𝑜 → (⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
2423opeq1d 3582 . . . . . . . . 9 (𝑧 = 1𝑜 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩)
2524eceq1d 6172 . . . . . . . 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 6874 . . . . . . . 8 1R = [⟨(1P +P 1P), 1P⟩] ~R
2725, 26syl6eqr 2106 . . . . . . 7 (𝑧 = 1𝑜 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
2827opeq1d 3582 . . . . . 6 (𝑧 = 1𝑜 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨1R, 0R⟩)
29 df-1 6954 . . . . . 6 1 = ⟨1R, 0R
3028, 29syl6eqr 2106 . . . . 5 (𝑧 = 1𝑜 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
3130eqeq1d 2064 . . . 4 (𝑧 = 1𝑜 → (⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1 ↔ 1 = 1))
3231rspcev 2673 . . 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 410 . 2 𝑧N ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
34 simplr 490 . . . . . . 7 (((𝑘𝑁𝑧N) ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → 𝑧N)
35 addclpi 6482 . . . . . . 7 ((𝑧N ∧ 1𝑜N) → (𝑧 +N 1𝑜) ∈ N)
3634, 10, 35sylancl 398 . . . . . 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 6980 . . . . . . . 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 107 . . . . . . . 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 5554 . . . . . . 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 2090 . . . . . 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 3576 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧 +N 1𝑜) → ⟨𝑣, 1𝑜⟩ = ⟨(𝑧 +N 1𝑜), 1𝑜⟩)
4342eceq1d 6172 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧 +N 1𝑜) → [⟨𝑣, 1𝑜⟩] ~Q = [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q )
4443breq2d 3803 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1𝑜) → (𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q ))
4544abbidv 2171 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1𝑜) → {𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q })
4643breq1d 3801 . . . . . . . . . . . . . 14 (𝑣 = (𝑧 +N 1𝑜) → ([⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢))
4746abbidv 2171 . . . . . . . . . . . . 13 (𝑣 = (𝑧 +N 1𝑜) → {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢})
4845, 47opeq12d 3584 . . . . . . . . . . . 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 5554 . . . . . . . . . . 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 3582 . . . . . . . . . 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 6172 . . . . . . . . 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 3582 . . . . . . . 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 2064 . . . . . . 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 2673 . . . . . 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 397 . . . . 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 112 . . . 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 2450 . . 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 3576 . . . . . . . . . . . . 13 (𝑣 = 𝑧 → ⟨𝑣, 1𝑜⟩ = ⟨𝑧, 1𝑜⟩)
5958eceq1d 6172 . . . . . . . . . . . 12 (𝑣 = 𝑧 → [⟨𝑣, 1𝑜⟩] ~Q = [⟨𝑧, 1𝑜⟩] ~Q )
6059breq2d 3803 . . . . . . . . . . 11 (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q ))
6160abbidv 2171 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑙𝑙 <Q [⟨𝑣, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑧, 1𝑜⟩] ~Q })
6259breq1d 3801 . . . . . . . . . . 11 (𝑣 = 𝑧 → ([⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢))
6362abbidv 2171 . . . . . . . . . 10 (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1𝑜⟩] ~Q <Q 𝑢})
6461, 63opeq12d 3584 . . . . . . . . 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 5554 . . . . . . . 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 3582 . . . . . . 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 6172 . . . . . 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 3582 . . . . 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 2064 . . . 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 2551 . . 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 154 . 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 7024 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 101   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  cop 3405   cint 3642   class class class wbr 3791  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   +N cpli 6428   ~Q ceq 6434  1Qc1q 6436   <Q cltq 6440  1Pc1p 6447   +P cpp 6448   ~R cer 6451  0Rc0r 6453  1Rc1r 6454  1c1 6947   + caddc 6949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-enr 6868  df-nr 6869  df-plr 6870  df-0r 6873  df-1r 6874  df-c 6952  df-1 6954  df-r 6956  df-add 6957
This theorem is referenced by:  axcaucvglemres  7030
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