Theorem uzsind 28671

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 23	. . . . . 6 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ ℤs)
2		zno 28648	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ No )
3		lesid 28004	. . . . . . . . 9 ⊢ (𝑀 ∈ No → 𝑀 ≤s 𝑀)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝑀 ∈ ℤs → 𝑀 ≤s 𝑀)
5		uzsind.5	. . . . . . . 8 ⊢ (𝑀 ∈ ℤs → 𝜓)
6	1, 4, 5	jca32 525	. . . . . . 7 ⊢ (𝑀 ∈ ℤs → (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
7		breq2 5111	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑀))
8		uzsind.1	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 644	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑀 ∧ 𝜓)))
10	9	elrab 3648	. . . . . . 7 ⊢ (𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
11	6, 10	sylibr 237	. . . . . 6 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
12		simpl 488	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ∈ ℤs)
13		simprl 783	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑘 ∈ ℤs)
14		simprrl 793	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ≤s 𝑘)
15		simprrr 794	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝜒)
16		id 23	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ ℤs)
17		1zs 28657	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 1s ∈ ℤs)
19	16, 18	zaddscld 28661	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ ℤs)
20	19	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ ℤs)
21	20	adantr 486	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → (𝑘 +s 1s ) ∈ ℤs)
22	2	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ∈ No )
23	19	znod 28649	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ No )
24	23	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ No )
25		zno 28648	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ No )
26	25	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 ∈ No )
27		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s 𝑘)
28	25	ltsp1d 28281	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 <s (𝑘 +s 1s ))
29	28	3ad2ant2 1152	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 <s (𝑘 +s 1s ))
30	22, 26, 24, 27, 29	leltstrd 28002	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 <s (𝑘 +s 1s ))
31	22, 24, 30	ltlesd 28010	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s (𝑘 +s 1s ))
32	31	adantr 486	. . . . . . . . 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 412	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → 𝜃)
35	21, 32, 34	jca32 525	. . . . . . . 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 1400	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
37		breq2 5111	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑘))
38		uzsind.2	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
39	37, 38	anbi12d 644	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑘 ∧ 𝜒)))
40	39	elrab 3648	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒)))
41	40	anbi2i 635	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) ↔ (𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))))
42		breq2 5111	. . . . . . . . 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 644	. . . . . . . 8 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
45	44	elrab 3648	. . . . . . 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 295	. . . . . 6 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) → (𝑘 +s 1s ) ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
47	1, 11, 46	peano5uzs 28670	. . . . 5 ⊢ (𝑀 ∈ ℤs → {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ⊆ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
48	47	sseld 3933	. . . 4 ⊢ (𝑀 ∈ ℤs → (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} → 𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}))
49		breq2 5111	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑀 ≤s 𝑤 ↔ 𝑀 ≤s 𝑁))
50	49	elrab 3648	. . . 4 ⊢ (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ↔ (𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁))
51		breq2 5111	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑁))
52		uzsind.4	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
53	51, 52	anbi12d 644	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑁 ∧ 𝜏)))
54	53	elrab 3648	. . . 4 ⊢ (𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
55	48, 50, 54	3imtr3g 298	. . 3 ⊢ (𝑀 ∈ ℤs → ((𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏))))
56	55	3impib 1134	. 2 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
57	56	simprrd 786	1 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3414   class class class wbr 5107  (class class class)co 7416   No csur 27877   <s clts 27878   ≤s cles 27981   1_s c1s 28072   +_s cadds 28225  ℤ_sczs 28644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-nadd 8657  df-no 27880  df-lts 27881  df-bday 27882  df-les 27982  df-slts 28024  df-cuts 28026  df-0s 28073  df-1s 28074  df-made 28093  df-old 28094  df-left 28096  df-right 28097  df-norec 28204  df-norec2 28215  df-adds 28226  df-negs 28287  df-subs 28288  df-n0s 28580  df-nns 28581  df-zs 28645

This theorem is used by: (None)