Theorem uzsind 28409

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 28386	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ No )
3		slerflex 27826	. . . . . . . . 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 5170	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑀))
8		uzsind.1	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 631	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑀 ∧ 𝜓)))
10	9	elrab 3708	. . . . . . 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 770	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑘 ∈ ℤs)
14		simprrl 780	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ≤s 𝑘)
15		simprrr 781	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝜒)
16		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ ℤs)
17		1zs 28395	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 1s ∈ ℤs)
19	16, 18	zaddscld 28399	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ ℤs)
20	19	3ad2ant2 1134	. . . . . . . . . 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 1133	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ∈ No )
23	19	znod 28387	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ No )
24	23	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ No )
25		zno 28386	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ No )
26	25	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 ∈ No )
27		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s 𝑘)
28	25	sltp1d 28066	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 <s (𝑘 +s 1s ))
29	28	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 <s (𝑘 +s 1s ))
30	22, 26, 24, 27, 29	slelttrd 27824	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 <s (𝑘 +s 1s ))
31	22, 24, 30	sltled 27832	. . . . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
37		breq2 5170	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑘))
38		uzsind.2	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
39	37, 38	anbi12d 631	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑘 ∧ 𝜒)))
40	39	elrab 3708	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒)))
41	40	anbi2i 622	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) ↔ (𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))))
42		breq2 5170	. . . . . . . . 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 631	. . . . . . . 8 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
45	44	elrab 3708	. . . . . . 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 28408	. . . . 5 ⊢ (𝑀 ∈ ℤs → {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ⊆ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
48	47	sseld 4007	. . . 4 ⊢ (𝑀 ∈ ℤs → (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} → 𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}))
49		breq2 5170	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑀 ≤s 𝑤 ↔ 𝑀 ≤s 𝑁))
50	49	elrab 3708	. . . 4 ⊢ (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ↔ (𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁))
51		breq2 5170	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑁))
52		uzsind.4	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
53	51, 52	anbi12d 631	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑁 ∧ 𝜏)))
54	53	elrab 3708	. . . 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 1116	. 2 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
57	56	simprrd 773	1 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {crab 3443   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   ≤s csle 27807   1_s c1s 27886   +_s cadds 28010  ℤ_sczs 28382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-n0s 28338  df-nns 28339  df-zs 28383

This theorem is referenced by: (None)