Theorem precsexlem6 28598

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 8646	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o ∅)))
4	3	sseq2d 3940	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅))))
5		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o 𝑗)))
7	6	sseq2d 3940	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑗))))
8		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 3940	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o 𝑘)) ↔ (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o suc 𝑗))))
11		nna0 8613	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6889	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝐿‘(𝐼 +o ∅)) = (𝐿‘𝐼))
13	12	eqimsscd 3965	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝐿‘𝐼) ⊆ (𝐿‘(𝐼 +o ∅)))
14		nnacl 8620	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4176	. . . . . . . . . . . . 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 28596	. . . . . . . . . . . . 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 3951	. . . . . . . . . . . 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 8615	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o suc 𝑗)) = (𝐿‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 3945	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐿‘(𝐼 +o 𝑗)) ⊆ (𝐿‘(𝐼 +o suc 𝑗)))
25		sstr2 3915	. . . . . . . . . 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 7910	. . . . . . 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 6885	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝐿‘(𝐼 +o 𝑘)) = (𝐿‘𝐽))
31	30	sseq2d 3940	. . . . . 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 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  {crab 3412  Vcvv 3450  ⦋csb 3846   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   ∘ ccom 5654  suc csuc 6364  ‘cfv 6538  (class class class)co 7420  ωcom 7877  1^st c1st 7999  2^nd c2nd 8000  reccrdg 8417   +_o coa 8473   <s clts 27998   0_s c0s 28191   1_s c1s 28192  Lcleft 28211  Rcright 28212   +_s cadds 28345   −_s csubs 28406   ·_s cmuls 28492   /_su cdivs 28573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480

This theorem is used by:  precsexlem10  28602