Theorem precsexlem6 28254

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 8693	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o ∅)))
4	3	sseq2d 4041	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅))))
5		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o 𝑗)))
7	6	sseq2d 4041	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗))))
8		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 4041	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
11		nna0 8660	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6924	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝐿‘(𝐼 +o ∅)) = (𝐿‘𝐼))
13	12	eqimsscd 4066	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅)))
14		nnacl 8667	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4201	. . . . . . . . . . . . 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 28252	. . . . . . . . . . . . 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 4062	. . . . . . . . . . . 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 8662	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o suc 𝑗)) = (𝐿‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 4050	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))
25		sstr2 4015	. . . . . . . . . 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 7938	. . . . . . 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 6920	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘𝐽))
31	30	sseq2d 4041	. . . . . 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 3161	. . . 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 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  {crab 3443  Vcvv 3488  ⦋csb 3921   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  {csn 4648  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249   ∘ ccom 5704  suc csuc 6397  ‘cfv 6573  (class class class)co 7448  ωcom 7903  1^st c1st 8028  2^nd c2nd 8029  reccrdg 8465   +_o coa 8519   <s cslt 27703   0_s c0s 27885   1_s c1s 27886   L cleft 27902   R cright 27903   +_s cadds 28010   -_s csubs 28070   ·_s cmuls 28150   /_su cdivs 28231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-oadd 8526

This theorem is referenced by:  precsexlem10  28258