Theorem precsexlem6 28531

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 8624	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6877	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o ∅)))
4	3	sseq2d 3962	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅))))
5		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6877	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o 𝑗)))
7	6	sseq2d 3962	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗))))
8		oveq2 7416	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6877	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3962	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
11		nna0 8591	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6877	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝐿‘(𝐼 +o ∅)) = (𝐿‘𝐼))
13	12	eqimsscd 3987	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅)))
14		nnacl 8598	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4123	. . . . . . . . . . . . 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 28529	. . . . . . . . . . . . 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 3973	. . . . . . . . . . . 12 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘suc (𝐼 +o 𝑗)))
21	14, 20	syl 18	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘suc (𝐼 +o 𝑗)))
22		nnasuc 8593	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6877	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o suc 𝑗)) = (𝐿‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 3967	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))
25		sstr2 3937	. . . . . . . . . 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 32	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗)) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
27	26	expcom 419	. . . . . . . 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 7893	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘))))
29	28	impcom 413	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)))
30		fveq2 6873	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘𝐽))
31	30	sseq2d 3962	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
32	29, 31	syl5ibcom 248	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
33	32	rexlimdva 3163	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
34	33	adantr 486	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
35	1, 34	sylbid 243	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝐿‘𝐼) ⊆ (𝐿‘𝐽)))
36	35	3impia 1135	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  {crab 3412  Vcvv 3450  ⦋csb 3846   ∪ cun 3896   ⊆ wss 3898  ∅c0 4278  {csn 4583  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   ∘ ccom 5651  suc csuc 6353  ‘cfv 6527  (class class class)co 7408  ωcom 7860  1^st c1st 7982  2^nd c2nd 7983  reccrdg 8395   +_o coa 8451   <s clts 27931   0_s c0s 28124   1_s c1s 28125   L cleft 28144   R cright 28145   +_s cadds 28278   -_s csubs 28339   ·_s cmuls 28425   /_su cdivs 28506

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8458

This theorem is used by:  precsexlem10  28535