Theorem recexprlemss1u 7748

Description: The upper cut of 𝐴 ·_P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7750. (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 7744	. . . . 5 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
3		df-imp 7581	. . . . . 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 7488	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
5	3, 4	genpelvu 7625	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
6	2, 5	mpdan 421	. . . 4 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
7	1	recexprlemelu 7735	. . . . . . . 8 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
8		ltrelnq 7477	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
9	8	brel 4726	. . . . . . . . . . . . 13 ⊢ (𝑦 <Q 𝑞 → (𝑦 ∈ Q ∧ 𝑞 ∈ Q))
10	9	simpld 112	. . . . . . . . . . . 12 ⊢ (𝑦 <Q 𝑞 → 𝑦 ∈ Q)
11		prop 7587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		elprnqu 7594	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
13	11, 12	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
14		ltmnqi 7515	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑧 ∈ Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
15	14	expcom 116	. . . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18		prltlu 7599	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
19	11, 18	syl3an1 1282	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
20	19	3com23 1211	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
21	20	3expia 1207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
22	21	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
23		ltmnqi 7515	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((*Q‘𝑦) <Q 𝑧 ∧ 𝑦 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧))
24	23	expcom 116	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
25	24	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
26		recidnq 7505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
27	26	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
28		mulcomnqg 7495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
29	27, 28	breq12d 4056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
30	25, 29	sylibd 149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
31	30	ancoms 268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
32	13, 31	sylan 283	. . . . . . . . . . . . . . . 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 335	. . . . . . . . . . . . . 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 7510	. . . . . . . . . . . . . . . 16 ⊢ <Q Or Q
36	35, 8	sotri 5077	. . . . . . . . . . . . . . 15 ⊢ ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
37	36	ancoms 268	. . . . . . . . . . . . . 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 367	. . . . . . . . . . . 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 254	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
43	42	exlimdv 1841	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
44	7, 43	biimtrid 152	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45		breq2 4047	. . . . . . . 8 ⊢ (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
46	45	biimprcd 160	. . . . . . 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 363	. . . . 5 ⊢ (𝐴 ∈ P → ((𝑧 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
49	48	rexlimdvv 2629	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
50	6, 49	sylbid 150	. . 3 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51		1pru 7668	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
52	51	abeq2i 2315	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
53	50, 52	imbitrrdi 162	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
54	53	ssrdv 3198	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  ∃wrex 2484   ⊆ wss 3165  ⟨cop 3635   class class class wbr 4043  ‘cfv 5270  (class class class)co 5943  1^st c1st 6223  2^nd c2nd 6224  Qcnq 7392  1_Qc1q 7393   ·_Q cmq 7395  *_Qcrq 7396   <_Q cltq 7397  Pcnp 7403  1_Pc1p 7404   ·_P cmp 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-inp 7578  df-i1p 7579  df-imp 7581

This theorem is referenced by:  recexprlemex  7749