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Theorem nn1suc 7714
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 6820 . . . . . 6 1 V
3 nn1suc.1 . . . . . 6 (x = 1 → (φψ))
42, 3sbcie 2791 . . . . 5 ([1 / x]φψ)
51, 4mpbir 134 . . . 4 [1 / x]φ
6 1nn 7706 . . . . . . 7 1
7 eleq1 2097 . . . . . . 7 (A = 1 → (A ℕ ↔ 1 ℕ))
86, 7mpbiri 157 . . . . . 6 (A = 1 → A ℕ)
9 nn1suc.4 . . . . . . 7 (x = A → (φθ))
109sbcieg 2789 . . . . . 6 (A ℕ → ([A / x]φθ))
118, 10syl 14 . . . . 5 (A = 1 → ([A / x]φθ))
12 dfsbcq 2760 . . . . 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 2562 . . . 4 ((A − 1) ℕ → y y = (A − 1))
17 eleq1 2097 . . . . . 6 (y = (A − 1) → (y ℕ ↔ (A − 1) ℕ))
1817pm5.32ri 428 . . . . 5 ((y y = (A − 1)) ↔ ((A − 1) y = (A − 1)))
19 nn1suc.6 . . . . . . 7 (y ℕ → χ)
2019adantr 261 . . . . . 6 ((y y = (A − 1)) → χ)
21 nnre 7702 . . . . . . . . 9 (y ℕ → y ℝ)
22 peano2re 6946 . . . . . . . . 9 (y ℝ → (y + 1) ℝ)
23 nn1suc.3 . . . . . . . . . 10 (x = (y + 1) → (φχ))
2423sbcieg 2789 . . . . . . . . 9 ((y + 1) ℝ → ([(y + 1) / x]φχ))
2521, 22, 243syl 17 . . . . . . . 8 (y ℕ → ([(y + 1) / x]φχ))
2625adantr 261 . . . . . . 7 ((y y = (A − 1)) → ([(y + 1) / x]φχ))
27 oveq1 5462 . . . . . . . . 9 (y = (A − 1) → (y + 1) = ((A − 1) + 1))
2827sbceq1d 2763 . . . . . . . 8 (y = (A − 1) → ([(y + 1) / x]φ[((A − 1) + 1) / x]φ))
2928adantl 262 . . . . . . 7 ((y y = (A − 1)) → ([(y + 1) / x]φ[((A − 1) + 1) / x]φ))
3026, 29bitr3d 179 . . . . . 6 ((y y = (A − 1)) → (χ[((A − 1) + 1) / x]φ))
3120, 30mpbid 135 . . . . 5 ((y y = (A − 1)) → [((A − 1) + 1) / x]φ)
3218, 31sylbir 125 . . . 4 (((A − 1) y = (A − 1)) → [((A − 1) + 1) / x]φ)
3316, 32exlimddv 1775 . . 3 ((A − 1) ℕ → [((A − 1) + 1) / x]φ)
34 nncn 7703 . . . . . 6 (A ℕ → A ℂ)
35 ax-1cn 6776 . . . . . 6 1
36 npcan 7017 . . . . . 6 ((A 1 ℂ) → ((A − 1) + 1) = A)
3734, 35, 36sylancl 392 . . . . 5 (A ℕ → ((A − 1) + 1) = A)
3837sbceq1d 2763 . . . 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 7713 . 2 (A ℕ → (A = 1 (A − 1) ℕ))
4215, 40, 41mpjaod 637 1 (A ℕ → θ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  [wsbc 2758  (class class class)co 5455  cc 6709  cr 6710  1c1 6712   + caddc 6714  cmin 6979  cn 7695
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-sub 6981  df-inn 7696
This theorem is referenced by: (None)
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