Theorem mulnqpru 7377

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 7209	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
2	1	adantl 275	. . . . . 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 7283	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elprnqu 7290	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
5	3, 4	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
6	5	ad2antrr 479	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
7		prop 7283	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
8		elprnqu 7290	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
9	7, 8	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
10	9	ad2antlr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
11		mulclnq 7184	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
12	6, 10, 11	syl2anc 408	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
13		simpr 109	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
14		recclnq 7200	. . . . . . 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 7191	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
17	16	adantl 275	. . . . . 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 5940	. . . . 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 7192	. . . . . . . 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 1216	. . . . . . 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 7201	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
22	21	oveq2d 5790	. . . . . . . 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 7197	. . . . . . . 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 2176	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = 𝐺)
27	26	breq1d 3939	. . . . 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 187	. . . 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 7293	. . . . . 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 281	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
31	30	ad2antrr 479	. . . 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 149	. . 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 7277	. . . . . . . . 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 7184	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
35	33, 34	genppreclu 7323	. . . . . . . 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 364	. . . . . . 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 83	. . . . . 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 255	. . . . 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 468	. . . 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 274	. . 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 45	. 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 7192	. . . . 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 1216	. . . 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 7191	. . . . . . 7 ⊢ (((*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
45	15, 10, 44	syl2anc 408	. . . . . 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 2172	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = 1Q)
48	47	oveq2d 5790	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49		mulidnq 7197	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
50	49	adantl 275	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
51	43, 48, 50	3eqtrd 2176	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = 𝑋)
52	51	eleq1d 2208	. 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 148	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 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088  1_Qc1q 7089   ·_Q cmq 7091  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099   ·_P cmp 7102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-mi 7114  df-lti 7115  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-imp 7277

This theorem is referenced by:  mullocprlem  7378  mulclpr  7380