Theorem uzsind 28406

Description: Induction on the upper surreal integers that start at 𝑀. (Contributed by Scott Fenton, 25-Jul-2025.)

Hypotheses

Ref	Expression
uzsind.1	⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
uzsind.2	⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
uzsind.3	⊢ (𝑗 = (𝑘 +s 1s ) → (𝜑 ↔ 𝜃))
uzsind.4	⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
uzsind.5	⊢ (𝑀 ∈ ℤs → 𝜓)
uzsind.6	⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝜒 → 𝜃))

Assertion

Ref	Expression
uzsind	⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

Distinct variable groups:   𝑗,𝑁   𝜓,𝑗   𝜒,𝑗   𝜃,𝑗   𝜏,𝑗   𝜑,𝑘   𝑗,𝑘,𝑀

Allowed substitution hints:   𝜑(𝑗)   𝜓(𝑘)   𝜒(𝑘)   𝜃(𝑘)   𝜏(𝑘)   𝑁(𝑘)

Proof of Theorem uzsind

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ ℤs)
2		zno 28383	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ No )
3		slerflex 27823	. . . . . . . . 9 ⊢ (𝑀 ∈ No → 𝑀 ≤s 𝑀)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℤs → 𝑀 ≤s 𝑀)
5		uzsind.5	. . . . . . . 8 ⊢ (𝑀 ∈ ℤs → 𝜓)
6	1, 4, 5	jca32 515	. . . . . . 7 ⊢ (𝑀 ∈ ℤs → (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
7		breq2 5152	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑀))
8		uzsind.1	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 632	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑀 ∧ 𝜓)))
10	9	elrab 3695	. . . . . . 7 ⊢ (𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
11	6, 10	sylibr 234	. . . . . 6 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
12		simpl 482	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ∈ ℤs)
13		simprl 771	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑘 ∈ ℤs)
14		simprrl 781	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ≤s 𝑘)
15		simprrr 782	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝜒)
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ ℤs)
17		1zs 28392	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 1s ∈ ℤs)
19	16, 18	zaddscld 28396	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ ℤs)
20	19	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ ℤs)
21	20	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → (𝑘 +s 1s ) ∈ ℤs)
22	2	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ∈ No )
23	19	znod 28384	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ No )
24	23	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ No )
25		zno 28383	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ No )
26	25	3ad2ant2 1133	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 ∈ No )
27		simp3 1137	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s 𝑘)
28	25	sltp1d 28063	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 <s (𝑘 +s 1s ))
29	28	3ad2ant2 1133	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 <s (𝑘 +s 1s ))
30	22, 26, 24, 27, 29	slelttrd 27821	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 <s (𝑘 +s 1s ))
31	22, 24, 30	sltled 27829	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s (𝑘 +s 1s ))
32	31	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → 𝑀 ≤s (𝑘 +s 1s ))
33		uzsind.6	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝜒 → 𝜃))
34	33	imp 406	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → 𝜃)
35	21, 32, 34	jca32 515	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
36	12, 13, 14, 15, 35	syl31anc 1372	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
37		breq2 5152	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑘))
38		uzsind.2	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
39	37, 38	anbi12d 632	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑘 ∧ 𝜒)))
40	39	elrab 3695	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒)))
41	40	anbi2i 623	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) ↔ (𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))))
42		breq2 5152	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s (𝑘 +s 1s )))
43		uzsind.3	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 +s 1s ) → (𝜑 ↔ 𝜃))
44	42, 43	anbi12d 632	. . . . . . . 8 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
45	44	elrab 3695	. . . . . . 7 ⊢ ((𝑘 +s 1s ) ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
46	36, 41, 45	3imtr4i 292	. . . . . 6 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) → (𝑘 +s 1s ) ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
47	1, 11, 46	peano5uzs 28405	. . . . 5 ⊢ (𝑀 ∈ ℤs → {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ⊆ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
48	47	sseld 3994	. . . 4 ⊢ (𝑀 ∈ ℤs → (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} → 𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}))
49		breq2 5152	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑀 ≤s 𝑤 ↔ 𝑀 ≤s 𝑁))
50	49	elrab 3695	. . . 4 ⊢ (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ↔ (𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁))
51		breq2 5152	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑁))
52		uzsind.4	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
53	51, 52	anbi12d 632	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑁 ∧ 𝜏)))
54	53	elrab 3695	. . . 4 ⊢ (𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
55	48, 50, 54	3imtr3g 295	. . 3 ⊢ (𝑀 ∈ ℤs → ((𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏))))
56	55	3impib 1115	. 2 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
57	56	simprrd 774	1 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   ≤s csle 27804   1_s c1s 27883   +_s cadds 28007  ℤ_sczs 28379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-n0s 28335  df-nns 28336  df-zs 28380

This theorem is referenced by: (None)