Theorem n0sind 28232

Description: Principle of Mathematical Induction (inference schema). Compare nnind 12211 and finds 7875. (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 28215	. . . 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 27745	. . . 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 28194	. 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 3450   ↦ cmpt 5191   “ cima 5644  (class class class)co 7390  ωcom 7845  reccrdg 8380   No csur 27558   0_s c0s 27741   1_s c1s 27742   +_s cadds 27873  ℕ_0scnn0s 28213

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-n0s 28215

This theorem is referenced by:  n0scut  28233  n0sge0  28237  n0s0suc  28241  n0addscl  28243  n0mulscl  28244  n0sbday  28251  n0s0m1  28259  n0subs  28260  n0p1nns  28267  dfnns2  28268  eucliddivs  28272  peano5uzs  28299  n0seo  28314  expscllem  28323  expadds  28327  expsne0  28328  expsgt0  28329  pw2recs  28330  pw2cut  28342  zs12bday  28350