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Theorem nn1suc 7933
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 7022 . . . . . 6 1 ∈ V
3 nn1suc.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
42, 3sbcie 2797 . . . . 5 ([1 / 𝑥]𝜑𝜓)
51, 4mpbir 134 . . . 4 [1 / 𝑥]𝜑
6 1nn 7925 . . . . . . 7 1 ∈ ℕ
7 eleq1 2100 . . . . . . 7 (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ))
86, 7mpbiri 157 . . . . . 6 (𝐴 = 1 → 𝐴 ∈ ℕ)
9 nn1suc.4 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜃))
109sbcieg 2795 . . . . . 6 (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑𝜃))
118, 10syl 14 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑𝜃))
12 dfsbcq 2766 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑[1 / 𝑥]𝜑))
1311, 12bitr3d 179 . . . 4 (𝐴 = 1 → (𝜃[1 / 𝑥]𝜑))
145, 13mpbiri 157 . . 3 (𝐴 = 1 → 𝜃)
1514a1i 9 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃))
16 elisset 2568 . . . 4 ((𝐴 − 1) ∈ ℕ → ∃𝑦 𝑦 = (𝐴 − 1))
17 eleq1 2100 . . . . . 6 (𝑦 = (𝐴 − 1) → (𝑦 ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ))
1817pm5.32ri 428 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) ↔ ((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)))
19 nn1suc.6 . . . . . . 7 (𝑦 ∈ ℕ → 𝜒)
2019adantr 261 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → 𝜒)
21 nnre 7921 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
22 peano2re 7149 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
23 nn1suc.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜒))
2423sbcieg 2795 . . . . . . . . 9 ((𝑦 + 1) ∈ ℝ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2521, 22, 243syl 17 . . . . . . . 8 (𝑦 ∈ ℕ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2625adantr 261 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
27 oveq1 5519 . . . . . . . . 9 (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1))
2827sbceq1d 2769 . . . . . . . 8 (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
2928adantl 262 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
3026, 29bitr3d 179 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → (𝜒[((𝐴 − 1) + 1) / 𝑥]𝜑))
3120, 30mpbid 135 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3218, 31sylbir 125 . . . 4 (((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3316, 32exlimddv 1778 . . 3 ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑)
34 nncn 7922 . . . . . 6 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
35 ax-1cn 6977 . . . . . 6 1 ∈ ℂ
36 npcan 7220 . . . . . 6 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴)
3734, 35, 36sylancl 392 . . . . 5 (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴)
3837sbceq1d 2769 . . . 4 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3938, 10bitrd 177 . . 3 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑𝜃))
4033, 39syl5ib 143 . 2 (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃))
41 nn1m1nn 7932 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
4215, 40, 41mpjaod 638 1 (𝐴 ∈ ℕ → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  [wsbc 2764  (class class class)co 5512  cc 6887  cr 6888  1c1 6890   + caddc 6892  cmin 7182  cn 7914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-inn 7915
This theorem is referenced by: (None)
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