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Theorem nn1suc 8009
Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
Hypotheses
Ref Expression
nn1suc.1 (𝑥 = 1 → (𝜑𝜓))
nn1suc.3 (𝑥 = (𝑦 + 1) → (𝜑𝜒))
nn1suc.4 (𝑥 = 𝐴 → (𝜑𝜃))
nn1suc.5 𝜓
nn1suc.6 (𝑦 ∈ ℕ → 𝜒)
Assertion
Ref Expression
nn1suc (𝐴 ∈ ℕ → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem nn1suc
StepHypRef Expression
1 nn1suc.5 . . . . 5 𝜓
2 1ex 7080 . . . . . 6 1 ∈ V
3 nn1suc.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
42, 3sbcie 2820 . . . . 5 ([1 / 𝑥]𝜑𝜓)
51, 4mpbir 138 . . . 4 [1 / 𝑥]𝜑
6 1nn 8001 . . . . . . 7 1 ∈ ℕ
7 eleq1 2116 . . . . . . 7 (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ))
86, 7mpbiri 161 . . . . . 6 (𝐴 = 1 → 𝐴 ∈ ℕ)
9 nn1suc.4 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜃))
109sbcieg 2818 . . . . . 6 (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑𝜃))
118, 10syl 14 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑𝜃))
12 dfsbcq 2789 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑[1 / 𝑥]𝜑))
1311, 12bitr3d 183 . . . 4 (𝐴 = 1 → (𝜃[1 / 𝑥]𝜑))
145, 13mpbiri 161 . . 3 (𝐴 = 1 → 𝜃)
1514a1i 9 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃))
16 elisset 2585 . . . 4 ((𝐴 − 1) ∈ ℕ → ∃𝑦 𝑦 = (𝐴 − 1))
17 eleq1 2116 . . . . . 6 (𝑦 = (𝐴 − 1) → (𝑦 ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ))
1817pm5.32ri 436 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) ↔ ((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)))
19 nn1suc.6 . . . . . . 7 (𝑦 ∈ ℕ → 𝜒)
2019adantr 265 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → 𝜒)
21 nnre 7997 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
22 peano2re 7210 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
23 nn1suc.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜒))
2423sbcieg 2818 . . . . . . . . 9 ((𝑦 + 1) ∈ ℝ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2521, 22, 243syl 17 . . . . . . . 8 (𝑦 ∈ ℕ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2625adantr 265 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
27 oveq1 5547 . . . . . . . . 9 (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1))
2827sbceq1d 2792 . . . . . . . 8 (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
2928adantl 266 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
3026, 29bitr3d 183 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → (𝜒[((𝐴 − 1) + 1) / 𝑥]𝜑))
3120, 30mpbid 139 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3218, 31sylbir 129 . . . 4 (((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3316, 32exlimddv 1794 . . 3 ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑)
34 nncn 7998 . . . . . 6 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
35 ax-1cn 7035 . . . . . 6 1 ∈ ℂ
36 npcan 7283 . . . . . 6 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴)
3734, 35, 36sylancl 398 . . . . 5 (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴)
3837sbceq1d 2792 . . . 4 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3938, 10bitrd 181 . . 3 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑𝜃))
4033, 39syl5ib 147 . 2 (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃))
41 nn1m1nn 8008 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
4215, 40, 41mpjaod 648 1 (𝐴 ∈ ℕ → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  [wsbc 2787  (class class class)co 5540  cc 6945  cr 6946  1c1 6948   + caddc 6950  cmin 7245  cn 7990
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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  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 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-inn 7991
This theorem is referenced by: (None)
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