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Theorem mulnqprl 7788

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 7621	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <_Q 𝑧 ↔ (𝑤 ·_Q 𝑦) <_Q (𝑤 ·_Q 𝑧)))
2	1	adantl 277	. . . . . 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 110	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
4		prop 7695	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7701	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
6	4, 5	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
7	6	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
8		prop 7695	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		elprnql 7701	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
10	8, 9	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
11	10	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
12		mulclnq 7596	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
13	7, 11, 12	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
14		recclnq 7612	. . . . . . 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 7603	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) = (𝑧 ·_Q 𝑦))
17	16	adantl 277	. . . . . 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 6192	. . . . 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 7604	. . . . . . . 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 1273	. . . . . . 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 7613	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
22	21	oveq2d 6034	. . . . . . . 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 7609	. . . . . . . 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 2268	. . . . . 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 4100	. . . . 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 188	. . . 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 7704	. . . . . 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 283	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → ((𝑋 ·_Q (_Q‘𝐻)) <_Q 𝐺 → (𝑋 ·_Q (_Q‘𝐻)) ∈ (1^st ‘𝐴)))
31	30	ad2antrr 488	. . . 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 150	. . 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 7689	. . . . . . . . 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 7596	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
35	33, 34	genpprecll 7734	. . . . . . . 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 367	. . . . . . 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 257	. . . . 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 477	. . . 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 276	. . 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 7604	. . . . 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 1273	. . . 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 7603	. . . . . . 7 ⊢ (((_Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((_Q‘𝐻) ·_Q 𝐻) = (𝐻 ·_Q (*_Q‘𝐻)))
45	15, 11, 44	syl2anc 411	. . . . . 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 2264	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘𝐻) ·_Q 𝐻) = 1_Q)
48	47	oveq2d 6034	. . . 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 7609	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·_Q 1_Q) = 𝑋)
50	49	adantl 277	. . . 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 2268	. . 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 2300	. 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 149	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 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 ⟨cop 3672 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 1^st c1st 6301 2^nd c2nd 6302 Qcnq 7500 1_Qc1q 7501 ·_Q cmq 7503 *_Qcrq 7504 <_Q cltq 7505 Pcnp 7511 ·_P cmp 7514

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-1o 6582 df-oadd 6586 df-omul 6587 df-er 6702 df-ec 6704 df-qs 6708 df-ni 7524 df-mi 7526 df-lti 7527 df-mpq 7565 df-enq 7567 df-nqqs 7568 df-mqqs 7570 df-1nqqs 7571 df-rq 7572 df-ltnqqs 7573 df-inp 7686 df-imp 7689

This theorem is referenced by: mullocprlem 7790 mulclpr 7792

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