Theorem recexprlemss1u 7098

Description: The upper cut of 𝐴 ·_P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7100. (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 7094	. . . . 5 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
3		df-imp 6931	. . . . . 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 6838	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
5	3, 4	genpelvu 6975	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
6	2, 5	mpdan 412	. . . 4 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
7	1	recexprlemelu 7085	. . . . . . . 8 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
8		ltrelnq 6827	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
9	8	brel 4448	. . . . . . . . . . . . 13 ⊢ (𝑦 <Q 𝑞 → (𝑦 ∈ Q ∧ 𝑞 ∈ Q))
10	9	simpld 110	. . . . . . . . . . . 12 ⊢ (𝑦 <Q 𝑞 → 𝑦 ∈ Q)
11		prop 6937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		elprnqu 6944	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
13	11, 12	sylan 277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
14		ltmnqi 6865	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑧 ∈ Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
15	14	expcom 114	. . . . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18		prltlu 6949	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
19	11, 18	syl3an1 1203	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
20	19	3com23 1145	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
21	20	3expia 1141	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
22	21	adantr 270	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
23		ltmnqi 6865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((*Q‘𝑦) <Q 𝑧 ∧ 𝑦 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧))
24	23	expcom 114	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
25	24	adantr 270	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
26		recidnq 6855	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
27	26	adantr 270	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
28		mulcomnqg 6845	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
29	27, 28	breq12d 3824	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
30	25, 29	sylibd 147	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
31	30	ancoms 264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
32	13, 31	sylan 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
33	22, 32	syld 44	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑦)))
34	17, 33	anim12d 328	. . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . . 16 ⊢ <Q Or Q
36	35, 8	sotri 4782	. . . . . . . . . . . . . . 15 ⊢ ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
37	36	ancoms 264	. . . . . . . . . . . . . 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 359	. . . . . . . . . . . 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 49	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞))))
42	41	impd 251	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
43	42	exlimdv 1742	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
44	7, 43	syl5bi 150	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45		breq2 3815	. . . . . . . 8 ⊢ (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
46	45	biimprcd 158	. . . . . . 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 355	. . . . 5 ⊢ (𝐴 ∈ P → ((𝑧 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
49	48	rexlimdvv 2489	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
50	6, 49	sylbid 148	. . 3 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51		1pru 7018	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
52	51	abeq2i 2193	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
53	50, 52	syl6ibr 160	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
54	53	ssrdv 3016	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  ∃wrex 2354   ⊆ wss 2984  ⟨cop 3425   class class class wbr 3811  ‘cfv 4969  (class class class)co 5591  1^st c1st 5844  2^nd c2nd 5845  Qcnq 6742  1_Qc1q 6743   ·_Q cmq 6745  *_Qcrq 6746   <_Q cltq 6747  Pcnp 6753  1_Pc1p 6754   ·_P cmp 6756

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-inp 6928  df-i1p 6929  df-imp 6931

This theorem is referenced by:  recexprlemex  7099