Theorem precsexlem7 27649

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 8634	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 ↔ ∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽))
2		oveq2 7414	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → (𝐼 +o 𝑘) = (𝐼 +o ∅))
3	2	fveq2d 6893	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o ∅)))
4	3	sseq2d 4014	. . . . . . . 8 ⊢ (𝑘 = ∅ → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅))))
5		oveq2 7414	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o 𝑗))
6	5	fveq2d 6893	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o 𝑗)))
7	6	sseq2d 4014	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑗))))
8		oveq2 7414	. . . . . . . . . 10 ⊢ (𝑘 = suc 𝑗 → (𝐼 +o 𝑘) = (𝐼 +o suc 𝑗))
9	8	fveq2d 6893	. . . . . . . . 9 ⊢ (𝑘 = suc 𝑗 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘(𝐼 +o suc 𝑗)))
10	9	sseq2d 4014	. . . . . . . 8 ⊢ (𝑘 = suc 𝑗 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o suc 𝑗))))
11		nna0 8601	. . . . . . . . . 10 ⊢ (𝐼 ∈ ω → (𝐼 +o ∅) = 𝐼)
12	11	fveq2d 6893	. . . . . . . . 9 ⊢ (𝐼 ∈ ω → (𝑅‘(𝐼 +o ∅)) = (𝑅‘𝐼))
13	12	eqimsscd 4039	. . . . . . . 8 ⊢ (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o ∅)))
14		nnacl 8608	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o 𝑗) ∈ ω)
15		ssun1 4172	. . . . . . . . . . . . 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 27647	. . . . . . . . . . . . 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 4035	. . . . . . . . . . . 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 8603	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝐼 +o suc 𝑗) = suc (𝐼 +o 𝑗))
23	22	fveq2d 6893	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o suc 𝑗)) = (𝑅‘suc (𝐼 +o 𝑗)))
24	21, 23	sseqtrrd 4023	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ω ∧ 𝑗 ∈ ω) → (𝑅‘(𝐼 +o 𝑗)) ⊆ (𝑅‘(𝐼 +o suc 𝑗)))
25		sstr2 3989	. . . . . . . . . 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 415	. . . . . . . 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 7888	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝐼 ∈ ω → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘))))
29	28	impcom 409	. . . . . 6 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → (𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)))
30		fveq2 6889	. . . . . . 7 ⊢ ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘(𝐼 +o 𝑘)) = (𝑅‘𝐽))
31	30	sseq2d 4014	. . . . . 6 ⊢ ((𝐼 +o 𝑘) = 𝐽 → ((𝑅‘𝐼) ⊆ (𝑅‘(𝐼 +o 𝑘)) ↔ (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
32	29, 31	syl5ibcom 244	. . . . 5 ⊢ ((𝐼 ∈ ω ∧ 𝑘 ∈ ω) → ((𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
33	32	rexlimdva 3156	. . . 4 ⊢ (𝐼 ∈ ω → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
34	33	adantr 482	. . 3 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (∃𝑘 ∈ ω (𝐼 +o 𝑘) = 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
35	1, 34	sylbid 239	. 2 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼 ⊆ 𝐽 → (𝑅‘𝐼) ⊆ (𝑅‘𝐽)))
36	35	3impia 1118	1 ⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  {crab 3433  Vcvv 3475  ⦋csb 3893   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  {csn 4628  ⟨cop 4634   class class class wbr 5148   ↦ cmpt 5231   ∘ ccom 5680  suc csuc 6364  ‘cfv 6541  (class class class)co 7406  ωcom 7852  1^st c1st 7970  2^nd c2nd 7971  reccrdg 8406   +_o coa 8460   <s cslt 27134   0_s c0s 27313   1_s c1s 27314   L cleft 27330   R cright 27331   +_s cadds 27433   -_s csubs 27485   ·_s cmuls 27552   /_su cdivs 27625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-oadd 8467

This theorem is referenced by:  precsexlem10  27652