Theorem precsexlem7 28172

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 8654	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o ∅)))
4	3	sseq2d 3996	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅))))
5		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o 𝑗)))
7	6	sseq2d 3996	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗))))
8		oveq2 7418	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3996	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
11		nna0 8621	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6885	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝑅‘(𝐼 +o ∅)) = (𝑅‘𝐼))
13	12	eqimsscd 4021	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅)))
14		nnacl 8628	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4158	. . . . . . . . . . . . 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 28170	. . . . . . . . . . . . 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 4007	. . . . . . . . . . . 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 8623	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o suc 𝑗)) = (𝑅‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 4001	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))
25		sstr2 3970	. . . . . . . . . 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 7899	. . . . . . 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 6881	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘𝐽))
31	30	sseq2d 3996	. . . . . 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 3142	. . . 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 1117	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2714  ∃wrex 3061  {crab 3420  Vcvv 3464  ⦋csb 3879   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612   class class class wbr 5124   ↦ cmpt 5206   ∘ ccom 5663  suc csuc 6359  ‘cfv 6536  (class class class)co 7410  ωcom 7866  1^st c1st 7991  2^nd c2nd 7992  reccrdg 8428   +_o coa 8482   <s cslt 27609   0_s c0s 27791   1_s c1s 27792   L cleft 27810   R cright 27811   +_s cadds 27923   -_s csubs 27983   ·_s cmuls 28066   /_su cdivs 28147

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-oadd 8489

This theorem is referenced by:  precsexlem10  28175