Theorem n0sind 28355

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12311 and finds 7936. (Contributed by Scott Fenton, 17-Mar-2025.)

Hypotheses

Ref	Expression
n0sind.1	⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))
n0sind.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
n0sind.3	⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))
n0sind.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
n0sind.5	⊢ 𝜓
n0sind.6	⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))

Assertion

Ref	Expression
n0sind	⊢ (𝐴 ∈ ℕ0s → 𝜏)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑦)   𝜃(𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem n0sind

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tru 1541	. 2 ⊢ ⊤
2		df-n0s 28338	. . . 4 ⊢ ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω)
3	2	a1i 11	. . 3 ⊢ (⊤ → ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω))
4		0sno 27889	. . . 4 ⊢ 0s ∈ No
5	4	a1i 11	. . 3 ⊢ (⊤ → 0s ∈ No )
6		n0sind.1	. . 3 ⊢ (𝑥 = 0s → (𝜑 ↔ 𝜓))
7		n0sind.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
8		n0sind.3	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))
9		n0sind.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
10		n0sind.5	. . . 4 ⊢ 𝜓
11	10	a1i 11	. . 3 ⊢ (⊤ → 𝜓)
12		n0sind.6	. . . 4 ⊢ (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))
13	12	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℕ0s) → (𝜒 → 𝜃))
14	3, 5, 6, 7, 8, 9, 11, 13	noseqinds 28317	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ ℕ0s) → 𝜏)
15	1, 14	mpan 689	1 ⊢ (𝐴 ∈ ℕ0s → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108  Vcvv 3488   ↦ cmpt 5249   “ cima 5703  (class class class)co 7448  ωcom 7903  reccrdg 8465   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-n0s 28338

This theorem is referenced by:  n0scut  28356  n0sge0  28359  n0s0suc  28363  n0addscl  28365  n0mulscl  28366  n0sbday  28372  n0s0m1  28377  n0subs  28378  n0p1nns  28379  dfnns2  28380  peano5uzs  28408  n0seo  28423  expscl  28431  expsne0  28432  expsgt0  28433  cutpw2  28435  pw2bday  28436  pw2cut  28438  zs12bday  28442