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Theorem mulnqprl 7224

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

**Assertion**
Ref	Expression
mulnqprl	⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) → 𝑋 ∈ (1^st ‘(𝐴 ·_P 𝐵))))

Proof of Theorem mulnqprl Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 7057	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
2	1	adantl 272	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
3		simpr 109	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
4		prop 7131	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7137	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
6	4, 5	sylan 278	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
7	6	ad2antrr 473	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
8		prop 7131	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		elprnql 7137	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
10	8, 9	sylan 278	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
11	10	ad2antlr 474	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
12		mulclnq 7032	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
13	7, 11, 12	syl2anc 404	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
14		recclnq 7048	. . . . . . 7 ⊢ (𝐻 ∈ Q → (*_Q‘𝐻) ∈ Q)
15	11, 14	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (*_Q‘𝐻) ∈ Q)
16		mulcomnqg 7039	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
17	16	adantl 272	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
18	2, 3, 13, 15, 17	caovord2d 5852	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) ↔ (𝑋 ·_Q (_Q‘𝐻)) <_Q ((𝐺 ·_Q 𝐻) ·_Q (_Q‘𝐻))))
19		mulassnqg 7040	. . . . . . . 8 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q ∧ (_Q‘𝐻) ∈ Q) → ((𝐺 ·_Q 𝐻) ·_Q (_Q‘𝐻)) = (𝐺 ·_Q (𝐻 ·_Q (*_Q‘𝐻))))
20	7, 11, 15, 19	syl3anc 1181	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·_Q 𝐻) ·_Q (_Q‘𝐻)) = (𝐺 ·_Q (𝐻 ·_Q (_Q‘𝐻))))
21		recidnq 7049	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
22	21	oveq2d 5706	. . . . . . . 8 ⊢ (𝐻 ∈ Q → (𝐺 ·_Q (𝐻 ·_Q (*_Q‘𝐻))) = (𝐺 ·_Q 1_Q))
23	11, 22	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q (𝐻 ·_Q (*_Q‘𝐻))) = (𝐺 ·_Q 1_Q))
24		mulidnq 7045	. . . . . . . 8 ⊢ (𝐺 ∈ Q → (𝐺 ·_Q 1_Q) = 𝐺)
25	7, 24	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q 1_Q) = 𝐺)
26	20, 23, 25	3eqtrd 2131	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·_Q 𝐻) ·_Q (*_Q‘𝐻)) = 𝐺)
27	26	breq2d 3879	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q ((𝐺 ·_Q 𝐻) ·_Q (_Q‘𝐻)) ↔ (𝑋 ·_Q (*_Q‘𝐻)) <_Q 𝐺))
28	18, 27	bitrd 187	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) ↔ (𝑋 ·_Q (*_Q‘𝐻)) <_Q 𝐺))
29		prcdnql 7140	. . . . . 6 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q 𝐺 → (𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴)))
30	4, 29	sylan 278	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q 𝐺 → (𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴)))
31	30	ad2antrr 473	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q 𝐺 → (𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴)))
32	28, 31	sylbid 149	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) → (𝑋 ·_Q (*_Q‘𝐻)) ∈ (1^st ‘𝐴)))
33		df-imp 7125	. . . . . . . . 9 ⊢ ·_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}⟩)
34		mulclnq 7032	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
35	33, 34	genpprecll 7170	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) ∧ 𝐻 ∈ (1^st ‘𝐵)) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
36	35	exp4b 360	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → ((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) → (𝐻 ∈ (1^st ‘𝐵) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))))
37	36	com34 83	. . . . . 6 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐻 ∈ (1^st ‘𝐵) → ((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))))
38	37	imp32 254	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) → ((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
39	38	adantlr 462	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) → ((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
40	39	adantr 271	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
41	32, 40	syld 45	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) → ((𝑋 ·_Q (*_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
42		mulassnqg 7040	. . . . 5 ⊢ ((𝑋 ∈ Q ∧ (_Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) = (𝑋 ·_Q ((*_Q‘𝐻) ·_Q 𝐻)))
43	3, 15, 11, 42	syl3anc 1181	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·_Q (_Q‘𝐻)) ·_Q 𝐻) = (𝑋 ·_Q ((_Q‘𝐻) ·_Q 𝐻)))
44		mulcomnqg 7039	. . . . . . 7 ⊢ (((_Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((_Q‘𝐻) ·_Q 𝐻) = (𝐻 ·_Q (*_Q‘𝐻)))
45	15, 11, 44	syl2anc 404	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((_Q‘𝐻) ·_Q 𝐻) = (𝐻 ·_Q (_Q‘𝐻)))
46	11, 21	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
47	45, 46	eqtrd 2127	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘𝐻) ·_Q 𝐻) = 1_Q)
48	47	oveq2d 5706	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q ((*_Q‘𝐻) ·_Q 𝐻)) = (𝑋 ·_Q 1_Q))
49		mulidnq 7045	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·_Q 1_Q) = 𝑋)
50	49	adantl 272	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q 1_Q) = 𝑋)
51	43, 48, 50	3eqtrd 2131	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·_Q (*_Q‘𝐻)) ·_Q 𝐻) = 𝑋)
52	51	eleq1d 2163	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝑋 ·_Q (*_Q‘𝐻)) ·_Q 𝐻) ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ 𝑋 ∈ (1^st ‘(𝐴 ·_P 𝐵))))
53	41, 52	sylibd 148	1 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 ·_Q 𝐻) → 𝑋 ∈ (1^st ‘(𝐴 ·_P 𝐵))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ⟨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 ·_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-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-mi 6962 df-lti 6963 df-mpq 7001 df-enq 7003 df-nqqs 7004 df-mqqs 7006 df-1nqqs 7007 df-rq 7008 df-ltnqqs 7009 df-inp 7122 df-imp 7125

This theorem is referenced by: mullocprlem 7226 mulclpr 7228

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