Theorem mulnqpru 6550

Description: Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)

Assertion

Ref	Expression
mulnqpru	⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(A ·P B))))

Proof of Theorem mulnqpru

Dummy variables v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 6385	. . . . . . 7 ⊢ ((y ∈ Q ∧ z ∈ Q ∧ w ∈ Q) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
2	1	adantl 262	. . . . . 6 ⊢ (((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) ∧ (y ∈ Q ∧ z ∈ Q ∧ w ∈ Q)) → (y <Q z ↔ (w ·Q y) <Q (w ·Q z)))
3		prop 6458	. . . . . . . . 9 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
4		elprnqu 6465	. . . . . . . . 9 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘A)) → 𝐺 ∈ Q)
5	3, 4	sylan 267	. . . . . . . 8 ⊢ ((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) → 𝐺 ∈ Q)
6	5	ad2antrr 457	. . . . . . 7 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
7		prop 6458	. . . . . . . . 9 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
8		elprnqu 6465	. . . . . . . . 9 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘B)) → 𝐻 ∈ Q)
9	7, 8	sylan 267	. . . . . . . 8 ⊢ ((B ∈ P ∧ 𝐻 ∈ (2nd ‘B)) → 𝐻 ∈ Q)
10	9	ad2antlr 458	. . . . . . 7 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
11		mulclnq 6360	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
12	6, 10, 11	syl2anc 391	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
13		simpr 103	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
14		recclnq 6376	. . . . . . 7 ⊢ (𝐻 ∈ Q → (*Q‘𝐻) ∈ Q)
15	10, 14	syl 14	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (*Q‘𝐻) ∈ Q)
16		mulcomnqg 6367	. . . . . . 7 ⊢ ((y ∈ Q ∧ z ∈ Q) → (y ·Q z) = (z ·Q y))
17	16	adantl 262	. . . . . 6 ⊢ (((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) ∧ (y ∈ Q ∧ z ∈ Q)) → (y ·Q z) = (z ·Q y))
18	2, 12, 13, 15, 17	caovord2d 5612	. . . . 5 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻))))
19		mulassnqg 6368	. . . . . . . 8 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q ∧ (*Q‘𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
20	6, 10, 15, 19	syl3anc 1134	. . . . . . 7 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
21		recidnq 6377	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
22	21	oveq2d 5471	. . . . . . . 8 ⊢ (𝐻 ∈ Q → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
23	10, 22	syl 14	. . . . . . 7 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
24		mulidnq 6373	. . . . . . . 8 ⊢ (𝐺 ∈ Q → (𝐺 ·Q 1Q) = 𝐺)
25	6, 24	syl 14	. . . . . . 7 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 1Q) = 𝐺)
26	20, 23, 25	3eqtrd 2073	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = 𝐺)
27	26	breq1d 3765	. . . . 5 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻)) ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
28	18, 27	bitrd 177	. . . 4 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
29		prcunqu 6468	. . . . . 6 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘A)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A)))
30	3, 29	sylan 267	. . . . 5 ⊢ ((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A)))
31	30	ad2antrr 457	. . . 4 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A)))
32	28, 31	sylbid 139	. . 3 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A)))
33		df-imp 6452	. . . . . . . . 9 ⊢ ·P = (w ∈ P, v ∈ P ↦ ⟨{x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (1st ‘w) ∧ z ∈ (1st ‘v) ∧ x = (y ·Q z))}, {x ∈ Q ∣ ∃y ∈ Q ∃z ∈ Q (y ∈ (2nd ‘w) ∧ z ∈ (2nd ‘v) ∧ x = (y ·Q z))}⟩)
34		mulclnq 6360	. . . . . . . . 9 ⊢ ((y ∈ Q ∧ z ∈ Q) → (y ·Q z) ∈ Q)
35	33, 34	genppreclu 6498	. . . . . . . 8 ⊢ ((A ∈ P ∧ B ∈ P) → (((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) ∧ 𝐻 ∈ (2nd ‘B)) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))
36	35	exp4b 349	. . . . . . 7 ⊢ (A ∈ P → (B ∈ P → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) → (𝐻 ∈ (2nd ‘B) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))))
37	36	com34 77	. . . . . 6 ⊢ (A ∈ P → (B ∈ P → (𝐻 ∈ (2nd ‘B) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))))
38	37	imp32 244	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))
39	38	adantlr 446	. . . 4 ⊢ (((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))
40	39	adantr 261	. . 3 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘A) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))
41	32, 40	syld 40	. 2 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B))))
42		mulassnqg 6368	. . . . 5 ⊢ ((𝑋 ∈ Q ∧ (*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
43	13, 15, 10, 42	syl3anc 1134	. . . 4 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
44		mulcomnqg 6367	. . . . . . 7 ⊢ (((*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
45	15, 10, 44	syl2anc 391	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
46	10, 21	syl 14	. . . . . 6 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
47	45, 46	eqtrd 2069	. . . . 5 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = 1Q)
48	47	oveq2d 5471	. . . 4 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49		mulidnq 6373	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
50	49	adantl 262	. . . 4 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
51	43, 48, 50	3eqtrd 2073	. . 3 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = 𝑋)
52	51	eleq1d 2103	. 2 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → (((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(A ·P B)) ↔ 𝑋 ∈ (2nd ‘(A ·P B))))
53	41, 52	sylibd 138	1 ⊢ ((((A ∈ P ∧ 𝐺 ∈ (2nd ‘A)) ∧ (B ∈ P ∧ 𝐻 ∈ (2nd ‘B))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(A ·P B))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264  1_Qc1q 6265   ·_Q cmq 6267  *_Qcrq 6268   <_Q cltq 6269  Pcnp 6275   ·_P cmp 6278

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-imp 6452

This theorem is referenced by:  mullocprlem  6551  mulclpr  6553