Theorem precsexlem7 28417

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 8621	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o ∅)))
4	3	sseq2d 3968	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅))))
5		oveq2 7420	. . . . . . . . . 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 3968	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗))))
8		oveq2 7420	. . . . . . . . . 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 3968	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
11		nna0 8588	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6885	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝑅‘(𝐼 +o ∅)) = (𝑅‘𝐼))
13	12	eqimsscd 3993	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅)))
14		nnacl 8595	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4130	. . . . . . . . . . . . 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 28415	. . . . . . . . . . . . 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 3979	. . . . . . . . . . . 12 ⊢ ((𝐼 +o 𝑗) ∈ ω → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘suc (𝐼 +o 𝑗)))
21	14, 20	syl 18	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘suc (𝐼 +o 𝑗)))
22		nnasuc 8590	. . . . . . . . . . . 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 3973	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))
25		sstr2 3943	. . . . . . . . . 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 32	. . . . . . . . 9 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗)) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
27	26	expcom 418	. . . . . . . 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 7893	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘))))
29	28	impcom 412	. . . . . 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 3968	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
32	29, 31	syl5ibcom 248	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
33	32	rexlimdva 3165	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
34	33	adantr 485	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
35	1, 34	sylbid 243	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
36	35	3impia 1134	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  {cab 2740  ∃wrex 3088  {crab 3415  Vcvv 3454  ⦋csb 3852   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4588  ⟨cop 4594   class class class wbr 5108   ↦ cmpt 5191   ∘ ccom 5664  suc csuc 6362  ‘cfv 6536  (class class class)co 7412  ωcom 7860  1^st c1st 7982  2^nd c2nd 7983  reccrdg 8394   +_o coa 8448   <s clts 27816   0_s c0s 28009   1_s c1s 28010   L cleft 28029   R cright 28030   +_s cadds 28163   -_s csubs 28224   ·_s cmuls 28310   /_su cdivs 28391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-oadd 8455

This theorem is used by:  precsexlem10  28420