Theorem n0sind 28248

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12164 and finds 7836. (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 1544	. 2 ⊢ ⊤
2		df-n0s 28231	. . . 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 27758	. . . 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 28210	. 2 ⊢ ((⊤ ∧ 𝐴 ∈ ℕ0s) → 𝜏)
15	1, 14	mpan 690	1 ⊢ (𝐴 ∈ ℕ0s → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  Vcvv 3438   ↦ cmpt 5176   “ cima 5626  (class class class)co 7353  ωcom 7806  reccrdg 8338   No csur 27567   0_s c0s 27754   1_s c1s 27755   +_s cadds 27889  ℕ_0scnn0s 28229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-no 27570  df-slt 27571  df-bday 27572  df-sslt 27710  df-scut 27712  df-0s 27756  df-n0s 28231

This theorem is referenced by:  n0scut  28249  n0sge0  28253  n0s0suc  28257  n0addscl  28259  n0mulscl  28260  n0sbday  28267  n0s0m1  28275  n0subs  28276  n0p1nns  28283  dfnns2  28284  eucliddivs  28288  peano5uzs  28315  n0seo  28331  expscllem  28340  expadds  28345  expsne0  28346  expsgt0  28347  pw2recs  28348  pw2cut  28366  zs12zodd  28377  zs12bday  28379