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Theorem nn1suc 8762
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 7784 . . . . . 6 1 ∈ V
3 nn1suc.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
42, 3sbcie 2946 . . . . 5 ([1 / 𝑥]𝜑𝜓)
51, 4mpbir 145 . . . 4 [1 / 𝑥]𝜑
6 1nn 8754 . . . . . . 7 1 ∈ ℕ
7 eleq1 2203 . . . . . . 7 (𝐴 = 1 → (𝐴 ∈ ℕ ↔ 1 ∈ ℕ))
86, 7mpbiri 167 . . . . . 6 (𝐴 = 1 → 𝐴 ∈ ℕ)
9 nn1suc.4 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜃))
109sbcieg 2944 . . . . . 6 (𝐴 ∈ ℕ → ([𝐴 / 𝑥]𝜑𝜃))
118, 10syl 14 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑𝜃))
12 dfsbcq 2914 . . . . 5 (𝐴 = 1 → ([𝐴 / 𝑥]𝜑[1 / 𝑥]𝜑))
1311, 12bitr3d 189 . . . 4 (𝐴 = 1 → (𝜃[1 / 𝑥]𝜑))
145, 13mpbiri 167 . . 3 (𝐴 = 1 → 𝜃)
1514a1i 9 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 → 𝜃))
16 elisset 2703 . . . 4 ((𝐴 − 1) ∈ ℕ → ∃𝑦 𝑦 = (𝐴 − 1))
17 eleq1 2203 . . . . . 6 (𝑦 = (𝐴 − 1) → (𝑦 ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ))
1817pm5.32ri 451 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) ↔ ((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)))
19 nn1suc.6 . . . . . . 7 (𝑦 ∈ ℕ → 𝜒)
2019adantr 274 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → 𝜒)
21 nnre 8750 . . . . . . . . 9 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
22 peano2re 7921 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ)
23 nn1suc.3 . . . . . . . . . 10 (𝑥 = (𝑦 + 1) → (𝜑𝜒))
2423sbcieg 2944 . . . . . . . . 9 ((𝑦 + 1) ∈ ℝ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2521, 22, 243syl 17 . . . . . . . 8 (𝑦 ∈ ℕ → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
2625adantr 274 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑𝜒))
27 oveq1 5788 . . . . . . . . 9 (𝑦 = (𝐴 − 1) → (𝑦 + 1) = ((𝐴 − 1) + 1))
2827sbceq1d 2917 . . . . . . . 8 (𝑦 = (𝐴 − 1) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
2928adantl 275 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → ([(𝑦 + 1) / 𝑥]𝜑[((𝐴 − 1) + 1) / 𝑥]𝜑))
3026, 29bitr3d 189 . . . . . 6 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → (𝜒[((𝐴 − 1) + 1) / 𝑥]𝜑))
3120, 30mpbid 146 . . . . 5 ((𝑦 ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3218, 31sylbir 134 . . . 4 (((𝐴 − 1) ∈ ℕ ∧ 𝑦 = (𝐴 − 1)) → [((𝐴 − 1) + 1) / 𝑥]𝜑)
3316, 32exlimddv 1871 . . 3 ((𝐴 − 1) ∈ ℕ → [((𝐴 − 1) + 1) / 𝑥]𝜑)
34 nncn 8751 . . . . . 6 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
35 ax-1cn 7736 . . . . . 6 1 ∈ ℂ
36 npcan 7994 . . . . . 6 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐴 − 1) + 1) = 𝐴)
3734, 35, 36sylancl 410 . . . . 5 (𝐴 ∈ ℕ → ((𝐴 − 1) + 1) = 𝐴)
3837sbceq1d 2917 . . . 4 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
3938, 10bitrd 187 . . 3 (𝐴 ∈ ℕ → ([((𝐴 − 1) + 1) / 𝑥]𝜑𝜃))
4033, 39syl5ib 153 . 2 (𝐴 ∈ ℕ → ((𝐴 − 1) ∈ ℕ → 𝜃))
41 nn1m1nn 8761 . 2 (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))
4215, 40, 41mpjaod 708 1 (𝐴 ∈ ℕ → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  [wsbc 2912  (class class class)co 5781  cc 7641  cr 7642  1c1 7644   + caddc 7646  cmin 7956  cn 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-sub 7958  df-inn 8744
This theorem is referenced by: (None)
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