Theorem recexprlemss1u 7292

Description: The upper cut of 𝐴 ·_P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7294. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemss1u	⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemss1u

Dummy variables 𝑞 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexpr.1	. . . . . 6 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
2	1	recexprlempr 7288	. . . . 5 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
3		df-imp 7125	. . . . . 6 ⊢ ·P = (𝑦 ∈ P, 𝑤 ∈ P ↦ ⟨{𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (1st ‘𝑦) ∧ 𝑔 ∈ (1st ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (2nd ‘𝑦) ∧ 𝑔 ∈ (2nd ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}⟩)
4		mulclnq 7032	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
5	3, 4	genpelvu 7169	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
6	2, 5	mpdan 413	. . . 4 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
7	1	recexprlemelu 7279	. . . . . . . 8 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
8		ltrelnq 7021	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
9	8	brel 4519	. . . . . . . . . . . . 13 ⊢ (𝑦 <Q 𝑞 → (𝑦 ∈ Q ∧ 𝑞 ∈ Q))
10	9	simpld 111	. . . . . . . . . . . 12 ⊢ (𝑦 <Q 𝑞 → 𝑦 ∈ Q)
11		prop 7131	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		elprnqu 7138	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
13	11, 12	sylan 278	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
14		ltmnqi 7059	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑧 ∈ Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
15	14	expcom 115	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ Q → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
16	13, 15	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
17	16	adantr 271	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18		prltlu 7143	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
19	11, 18	syl3an1 1214	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
20	19	3com23 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
21	20	3expia 1148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
22	21	adantr 271	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
23		ltmnqi 7059	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((*Q‘𝑦) <Q 𝑧 ∧ 𝑦 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧))
24	23	expcom 115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
25	24	adantr 271	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
26		recidnq 7049	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
27	26	adantr 271	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
28		mulcomnqg 7039	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
29	27, 28	breq12d 3880	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
30	25, 29	sylibd 148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
31	30	ancoms 265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
32	13, 31	sylan 278	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
33	22, 32	syld 45	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑦)))
34	17, 33	anim12d 329	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦))))
35		ltsonq 7054	. . . . . . . . . . . . . . . 16 ⊢ <Q Or Q
36	35, 8	sotri 4860	. . . . . . . . . . . . . . 15 ⊢ ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
37	36	ancoms 265	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦)) → 1Q <Q (𝑧 ·Q 𝑞))
38	34, 37	syl6 33	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
39	38	exp4b 360	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 ∈ Q → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
40	10, 39	syl5 32	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
41	40	pm2.43d 50	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞))))
42	41	impd 252	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
43	42	exlimdv 1754	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
44	7, 43	syl5bi 151	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45		breq2 3871	. . . . . . . 8 ⊢ (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
46	45	biimprcd 159	. . . . . . 7 ⊢ (1Q <Q (𝑧 ·Q 𝑞) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
47	44, 46	syl6 33	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
48	47	expimpd 356	. . . . 5 ⊢ (𝐴 ∈ P → ((𝑧 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
49	48	rexlimdvv 2509	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
50	6, 49	sylbid 149	. . 3 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51		1pru 7212	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
52	51	abeq2i 2205	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
53	50, 52	syl6ibr 161	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
54	53	ssrdv 3045	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081  ∃wrex 2371   ⊆ wss 3013  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936  1_Qc1q 6937   ·_Q cmq 6939  *_Qcrq 6940   <_Q cltq 6941  Pcnp 6947  1_Pc1p 6948   ·_P cmp 6950

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-inp 7122  df-i1p 7123  df-imp 7125

This theorem is referenced by:  recexprlemex  7293