Theorem precsexlem7 28193

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 8562	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6833	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o ∅)))
4	3	sseq2d 3949	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅))))
5		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6833	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o 𝑗)))
7	6	sseq2d 3949	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗))))
8		oveq2 7364	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6833	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3949	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
11		nna0 8529	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6833	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝑅‘(𝐼 +o ∅)) = (𝑅‘𝐼))
13	12	eqimsscd 3974	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅)))
14		nnacl 8536	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4109	. . . . . . . . . . . . 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 28191	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . 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 8531	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6833	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o suc 𝑗)) = (𝑅‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 3954	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))
25		sstr2 3924	. . . . . . . . . 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 7838	. . . . . . 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 6829	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘𝐽))
31	30	sseq2d 3949	. . . . . 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 3136	. . . 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 1542   ∈ wcel 2114  {cab 2713  ∃wrex 3059  {crab 3387  Vcvv 3427  ⦋csb 3833   ∪ cun 3883   ⊆ wss 3885  ∅c0 4263  {csn 4557  ⟨cop 4563   class class class wbr 5074   ↦ cmpt 5155   ∘ ccom 5624  suc csuc 6314  ‘cfv 6487  (class class class)co 7356  ωcom 7806  1^st c1st 7929  2^nd c2nd 7930  reccrdg 8337   +_o coa 8391   <s clts 27592   0_s c0s 27785   1_s c1s 27786   L cleft 27805   R cright 27806   +_s cadds 27939   -_s csubs 28000   ·_s cmuls 28086   /_su cdivs 28167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-oadd 8398

This theorem is referenced by:  precsexlem10  28196