Theorem precsexlem6 28307

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 8609	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7406	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6873	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o ∅)))
4	3	sseq2d 3970	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅))))
5		oveq2 7406	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6873	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o 𝑗)))
7	6	sseq2d 3970	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗))))
8		oveq2 7406	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6873	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3970	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
11		nna0 8576	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6873	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝐿‘(𝐼 +o ∅)) = (𝐿‘𝐼))
13	12	eqimsscd 3995	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅)))
14		nnacl 8583	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4132	. . . . . . . . . . . . 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 28305	. . . . . . . . . . . . 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 3981	. . . . . . . . . . . 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 8578	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6873	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o suc 𝑗)) = (𝐿‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 3975	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))
25		sstr2 3945	. . . . . . . . . 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 417	. . . . . . . 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 7881	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘))))
29	28	impcom 411	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)))
30		fveq2 6869	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘𝐽))
31	30	sseq2d 3970	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
32	29, 31	syl5ibcom 247	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
33	32	rexlimdva 3165	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
34	33	adantr 484	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
35	1, 34	sylbid 242	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
36	35	3impia 1131	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘𝐽))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  {cab 2742  ∃wrex 3088  {crab 3416  Vcvv 3456  ⦋csb 3854   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183   ∘ ccom 5653  suc csuc 6350  ‘cfv 6523  (class class class)co 7398  ωcom 7848  1^st c1st 7970  2^nd c2nd 7971  reccrdg 8382   +_o coa 8436   <s clts 27707   0_s c0s 27900   1_s c1s 27901   L cleft 27920   R cright 27921   +_s cadds 28054   -_s csubs 28115   ·_s cmuls 28201   /_su cdivs 28282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-oadd 8443

This theorem is referenced by:  precsexlem10  28311