Theorem precsexlem6 28166

Description: Lemma for surreal reciprocal. Show that 𝐿 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
precsexlem6	⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘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 precsexlem6

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

Step	Hyp	Ref	Expression
1		nnawordex 8649	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o ∅)))
4	3	sseq2d 3991	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅))))
5		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o 𝑗)))
7	6	sseq2d 3991	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗))))
8		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3991	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
11		nna0 8616	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6880	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝐿‘(𝐼 +o ∅)) = (𝐿‘𝐼))
13	12	eqimsscd 4016	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅)))
14		nnacl 8623	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4153	. . . . . . . . . . . . 13 ⊢ (𝐿‘(𝐼 +o 𝑗)) ⊆ ((𝐿‘(𝐼 +o 𝑗)) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)}))
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	precsexlem4 28164	. . . . . . . . . . . . 13 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝐿‘suc (𝐼 +o 𝑗)) = ((𝐿‘(𝐼 +o 𝑗)) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘(𝐼 +o 𝑗))𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
20	15, 19	sseqtrrid 4002	. . . . . . . . . . . 12 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘suc (𝐼 +o 𝑗)))
21	14, 20	syl 17	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘suc (𝐼 +o 𝑗)))
22		nnasuc 8618	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o suc 𝑗)) = (𝐿‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 3996	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))
25		sstr2 3965	. . . . . . . . . 10 ⊢ ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗)) → ((𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
26	24, 25	syl5com 31	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗)) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
27	26	expcom 413	. . . . . . . 8 ⊢ (𝑗 ∈ ω → (𝐼 ∈ ω → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗)) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))))
28	4, 7, 10, 13, 27	finds2 7894	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘))))
29	28	impcom 407	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)))
30		fveq2 6876	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘𝐽))
31	30	sseq2d 3991	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
32	29, 31	syl5ibcom 245	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
33	32	rexlimdva 3141	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
34	33	adantr 480	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
35	1, 34	sylbid 240	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
36	35	3impia 1117	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  {cab 2713  ∃wrex 3060  {crab 3415  Vcvv 3459  ⦋csb 3874   ∪ cun 3924   ⊆ wss 3926  ∅c0 4308  {csn 4601  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   ∘ ccom 5658  suc csuc 6354  ‘cfv 6531  (class class class)co 7405  ωcom 7861  1^st c1st 7986  2^nd c2nd 7987  reccrdg 8423   +_o coa 8477   <s cslt 27604   0_s c0s 27786   1_s c1s 27787   L cleft 27805   R cright 27806   +_s cadds 27918   -_s csubs 27978   ·_s cmuls 28061   /_su cdivs 28142

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-oadd 8484

This theorem is referenced by:  precsexlem10  28170