Theorem uzsind 28422

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 28399	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ No )
3		lesid 27756	. . . . . . . . 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 520	. . . . . . 7 ⊢ (𝑀 ∈ ℤs → (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
7		breq2 5083	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑀))
8		uzsind.1	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 638	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑀 ∧ 𝜓)))
10	9	elrab 3636	. . . . . . 7 ⊢ (𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
11	6, 10	sylibr 235	. . . . . 6 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
12		simpl 483	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ∈ ℤs)
13		simprl 776	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑘 ∈ ℤs)
14		simprrl 786	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ≤s 𝑘)
15		simprrr 787	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝜒)
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ ℤs)
17		1zs 28408	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 1s ∈ ℤs)
19	16, 18	zaddscld 28412	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ ℤs)
20	19	3ad2ant2 1140	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ ℤs)
21	20	adantr 481	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → (𝑘 +s 1s ) ∈ ℤs)
22	2	3ad2ant1 1139	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ∈ No )
23	19	znod 28400	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ No )
24	23	3ad2ant2 1140	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ No )
25		zno 28399	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ No )
26	25	3ad2ant2 1140	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 ∈ No )
27		simp3 1144	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s 𝑘)
28	25	ltsp1d 28032	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 <s (𝑘 +s 1s ))
29	28	3ad2ant2 1140	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 <s (𝑘 +s 1s ))
30	22, 26, 24, 27, 29	leltstrd 27754	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 <s (𝑘 +s 1s ))
31	22, 24, 30	ltlesd 27762	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s (𝑘 +s 1s ))
32	31	adantr 481	. . . . . . . . 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 407	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → 𝜃)
35	21, 32, 34	jca32 520	. . . . . . . 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 1381	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
37		breq2 5083	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑘))
38		uzsind.2	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
39	37, 38	anbi12d 638	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑘 ∧ 𝜒)))
40	39	elrab 3636	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒)))
41	40	anbi2i 629	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) ↔ (𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))))
42		breq2 5083	. . . . . . . . 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 638	. . . . . . . 8 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
45	44	elrab 3636	. . . . . . 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 293	. . . . . 6 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) → (𝑘 +s 1s ) ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
47	1, 11, 46	peano5uzs 28421	. . . . 5 ⊢ (𝑀 ∈ ℤs → {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ⊆ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
48	47	sseld 3921	. . . 4 ⊢ (𝑀 ∈ ℤs → (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} → 𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}))
49		breq2 5083	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑀 ≤s 𝑤 ↔ 𝑀 ≤s 𝑁))
50	49	elrab 3636	. . . 4 ⊢ (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ↔ (𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁))
51		breq2 5083	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑁))
52		uzsind.4	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
53	51, 52	anbi12d 638	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑁 ∧ 𝜏)))
54	53	elrab 3636	. . . 4 ⊢ (𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
55	48, 50, 54	3imtr3g 296	. . 3 ⊢ (𝑀 ∈ ℤs → ((𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏))))
56	55	3impib 1122	. 2 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
57	56	simprrd 779	1 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {crab 3392   class class class wbr 5079  (class class class)co 7363   No csur 27628   <s clts 27629   ≤s cles 27733   1_s c1s 27823   +_s cadds 27976  ℤ_sczs 28395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-n0s 28331  df-nns 28332  df-zs 28396

This theorem is referenced by: (None)