Theorem precsexlem7 28237

Description: Lemma for surreal reciprocal. Show that 𝑅 is non-strictly increasing in its argument. (Contributed by Scott Fenton, 15-Mar-2025.)

Hypotheses

Ref	Expression
precsexlem.1	⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
precsexlem.2	⊢ 𝐿 = (1st ∘ 𝐹)
precsexlem.3	⊢ 𝑅 = (2nd ∘ 𝐹)

Assertion

Ref	Expression
precsexlem7	⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))

Distinct variable groups:   𝐴,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥_𝐿,𝑥_𝑅,𝑦_𝑅   𝐹,𝑙,𝑝   𝐼,𝑎,𝑙,𝑝,𝑟,𝑥,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅   𝐿,𝑎,𝑙,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿   𝑅,𝑎,𝑙,𝑟,𝑥_𝐿,𝑥_𝑅,𝑦_𝑅

Allowed substitution hints:   𝐴(𝑦_𝐿)   𝑅(𝑥,𝑝,𝑦_𝐿)   𝐹(𝑥,𝑟,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅)   𝐽(𝑥,𝑟,𝑝,𝑎,𝑙,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅)   𝐿(𝑥,𝑟,𝑝,𝑦_𝑅)

Proof of Theorem precsexlem7

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnawordex 8675	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o ∅)))
4	3	sseq2d 4016	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅))))
5		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o 𝑗)))
7	6	sseq2d 4016	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗))))
8		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 4016	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
11		nna0 8642	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6910	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝑅‘(𝐼 +o ∅)) = (𝑅‘𝐼))
13	12	eqimsscd 4041	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅)))
14		nnacl 8649	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4178	. . . . . . . . . . . . 13 ⊢ (𝑅‘(𝐼 +o 𝑗)) ⊆ ((𝑅‘(𝐼 +o 𝑗)) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))
16		precsexlem.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
17		precsexlem.2	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (1st ∘ 𝐹)
18		precsexlem.3	. . . . . . . . . . . . . 14 ⊢ 𝑅 = (2nd ∘ 𝐹)
19	16, 17, 18	precsexlem5 28235	. . . . . . . . . . . . 13 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝑅‘suc (𝐼 +o 𝑗)) = ((𝑅‘(𝐼 +o 𝑗)) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
20	15, 19	sseqtrrid 4027	. . . . . . . . . . . 12 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘suc (𝐼 +o 𝑗)))
21	14, 20	syl 17	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘suc (𝐼 +o 𝑗)))
22		nnasuc 8644	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6910	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o suc 𝑗)) = (𝑅‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 4021	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))
25		sstr2 3990	. . . . . . . . . 10 ⊢ ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗)) → ((𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
26	24, 25	syl5com 31	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗)) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
27	26	expcom 413	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝐼 ∈ ω → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗)) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))))
28	4, 7, 10, 13, 27	finds2 7920	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘))))
29	28	impcom 407	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)))
30		fveq2 6906	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘𝐽))
31	30	sseq2d 4016	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
32	29, 31	syl5ibcom 245	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
33	32	rexlimdva 3155	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
34	33	adantr 480	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
35	1, 34	sylbid 240	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
36	35	3impia 1118	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  {crab 3436  Vcvv 3480  ⦋csb 3899   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  ⟨cop 4632   class class class wbr 5143   ↦ cmpt 5225   ∘ ccom 5689  suc csuc 6386  ‘cfv 6561  (class class class)co 7431  ωcom 7887  1^st c1st 8012  2^nd c2nd 8013  reccrdg 8449   +_o coa 8503   <s cslt 27685   0_s c0s 27867   1_s c1s 27868   L cleft 27884   R cright 27885   +_s cadds 27992   -_s csubs 28052   ·_s cmuls 28132   /_su cdivs 28213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510

This theorem is referenced by:  precsexlem10  28240