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Theorem mulnqprl 7369

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 7202	. . . . . . 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 ∈ (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 7276	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7282	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
6	4, 5	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
7	6	ad2antrr 479	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
8		prop 7276	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		elprnql 7282	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
10	8, 9	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
11	10	ad2antlr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
12		mulclnq 7177	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
13	7, 11, 12	syl2anc 408	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
14		recclnq 7193	. . . . . . 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 7184	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
17	16	adantl 275	. . . . . 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 5933	. . . . 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 7185	. . . . . . . 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 1216	. . . . . . 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 7194	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
22	21	oveq2d 5783	. . . . . . . 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 7190	. . . . . . . 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 2174	. . . . . 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 3936	. . . . 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 7285	. . . . . 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 281	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q 𝐺 → (𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴)))
31	30	ad2antrr 479	. . . 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 7270	. . . . . . . . 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 7177	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
35	33, 34	genpprecll 7315	. . . . . . . 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 364	. . . . . . 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 255	. . . . 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 468	. . . 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 274	. . 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 7185	. . . . 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 1216	. . . 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 7184	. . . . . . 7 ⊢ (((_Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((_Q‘𝐻) ·_Q 𝐻) = (𝐻 ·_Q (*_Q‘𝐻)))
45	15, 11, 44	syl2anc 408	. . . . . 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 2170	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘𝐻) ·_Q 𝐻) = 1_Q)
48	47	oveq2d 5783	. . . 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 7190	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·_Q 1_Q) = 𝑋)
50	49	adantl 275	. . . 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 2174	. . 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 2206	. 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 962 = wceq 1331 ∈ wcel 1480 ⟨cop 3525 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 1^st c1st 6029 2^nd c2nd 6030 Qcnq 7081 1_Qc1q 7082 ·_Q cmq 7084 *_Qcrq 7085 <_Q cltq 7086 Pcnp 7092 ·_P cmp 7095

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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497

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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-mi 7107 df-lti 7108 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-inp 7267 df-imp 7270

This theorem is referenced by: mullocprlem 7371 mulclpr 7373

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