Theorem uzsind 28574

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 28551	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤs → 𝑀 ∈ No )
3		lesid 27907	. . . . . . . . 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 524	. . . . . . 7 ⊢ (𝑀 ∈ ℤs → (𝑀 ∈ ℤs ∧ (𝑀 ≤s 𝑀 ∧ 𝜓)))
7		breq2 5112	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑀))
8		uzsind.1	. . . . . . . . 9 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 643	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑀 ∧ 𝜓)))
10	9	elrab 3649	. . . . . . 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 487	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ∈ ℤs)
13		simprl 782	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑘 ∈ ℤs)
14		simprrl 792	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝑀 ≤s 𝑘)
15		simprrr 793	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → 𝜒)
16		id 23	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ ℤs)
17		1zs 28560	. . . . . . . . . . . . 13 ⊢ 1s ∈ ℤs
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → 1s ∈ ℤs)
19	16, 18	zaddscld 28564	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ ℤs)
20	19	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ ℤs)
21	20	adantr 485	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → (𝑘 +s 1s ) ∈ ℤs)
22	2	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ∈ No )
23	19	znod 28552	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤs → (𝑘 +s 1s ) ∈ No )
24	23	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝑘 +s 1s ) ∈ No )
25		zno 28551	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 ∈ No )
26	25	3ad2ant2 1150	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 ∈ No )
27		simp3 1154	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s 𝑘)
28	25	ltsp1d 28184	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤs → 𝑘 <s (𝑘 +s 1s ))
29	28	3ad2ant2 1150	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑘 <s (𝑘 +s 1s ))
30	22, 26, 24, 27, 29	leltstrd 27905	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 <s (𝑘 +s 1s ))
31	22, 24, 30	ltlesd 27913	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → 𝑀 ≤s (𝑘 +s 1s ))
32	31	adantr 485	. . . . . . . . 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 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) ∧ 𝜒) → 𝜃)
35	21, 32, 34	jca32 524	. . . . . . . 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 1398	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))) → ((𝑘 +s 1s ) ∈ ℤs ∧ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
37		breq2 5112	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑘))
38		uzsind.2	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))
39	37, 38	anbi12d 643	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑘 ∧ 𝜒)))
40	39	elrab 3649	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)} ↔ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒)))
41	40	anbi2i 634	. . . . . . 7 ⊢ ((𝑀 ∈ ℤs ∧ 𝑘 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}) ↔ (𝑀 ∈ ℤs ∧ (𝑘 ∈ ℤs ∧ (𝑀 ≤s 𝑘 ∧ 𝜒))))
42		breq2 5112	. . . . . . . . 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 643	. . . . . . . 8 ⊢ (𝑗 = (𝑘 +s 1s ) → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s (𝑘 +s 1s ) ∧ 𝜃)))
45	44	elrab 3649	. . . . . . 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 28573	. . . . 5 ⊢ (𝑀 ∈ ℤs → {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ⊆ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)})
48	47	sseld 3935	. . . 4 ⊢ (𝑀 ∈ ℤs → (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} → 𝑁 ∈ {𝑗 ∈ ℤs ∣ (𝑀 ≤s 𝑗 ∧ 𝜑)}))
49		breq2 5112	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑀 ≤s 𝑤 ↔ 𝑀 ≤s 𝑁))
50	49	elrab 3649	. . . 4 ⊢ (𝑁 ∈ {𝑤 ∈ ℤs ∣ 𝑀 ≤s 𝑤} ↔ (𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁))
51		breq2 5112	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝑀 ≤s 𝑗 ↔ 𝑀 ≤s 𝑁))
52		uzsind.4	. . . . . 6 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))
53	51, 52	anbi12d 643	. . . . 5 ⊢ (𝑗 = 𝑁 → ((𝑀 ≤s 𝑗 ∧ 𝜑) ↔ (𝑀 ≤s 𝑁 ∧ 𝜏)))
54	53	elrab 3649	. . . 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 1132	. 2 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → (𝑁 ∈ ℤs ∧ (𝑀 ≤s 𝑁 ∧ 𝜏)))
57	56	simprrd 785	1 ⊢ ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {crab 3414   class class class wbr 5108  (class class class)co 7410   No csur 27780   <s clts 27781   ≤s cles 27884   1_s c1s 27975   +_s cadds 28128  ℤ_sczs 28547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-nadd 8651  df-no 27783  df-lts 27784  df-bday 27785  df-les 27885  df-slts 27927  df-cuts 27929  df-0s 27976  df-1s 27977  df-made 27996  df-old 27997  df-left 27999  df-right 28000  df-norec 28107  df-norec2 28118  df-adds 28129  df-negs 28190  df-subs 28191  df-n0s 28483  df-nns 28484  df-zs 28548

This theorem is referenced by: (None)