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Theorem mulnqprl 7763

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 7596	. . . . . . 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 7670	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7676	. . . . . . . . 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 7670	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		elprnql 7676	. . . . . . . . 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 7571	. . . . . . 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 7587	. . . . . . 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 7578	. . . . . . 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 6181	. . . . 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 7579	. . . . . . . 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 1271	. . . . . . 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 7588	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
22	21	oveq2d 6023	. . . . . . . 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 7584	. . . . . . . 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 2266	. . . . . 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 4095	. . . . 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 7679	. . . . . 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 7664	. . . . . . . . 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 7571	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
35	33, 34	genpprecll 7709	. . . . . . . 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 7579	. . . . 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 1271	. . . 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 7578	. . . . . . 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 2262	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘𝐻) ·_Q 𝐻) = 1_Q)
48	47	oveq2d 6023	. . . 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 7584	. . . . 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 2266	. . 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 2298	. 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 1002 = wceq 1395 ∈ wcel 2200 ⟨cop 3669 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 1^st c1st 6290 2^nd c2nd 6291 Qcnq 7475 1_Qc1q 7476 ·_Q cmq 7478 *_Qcrq 7479 <_Q cltq 7480 Pcnp 7486 ·_P cmp 7489

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-1o 6568 df-oadd 6572 df-omul 6573 df-er 6688 df-ec 6690 df-qs 6694 df-ni 7499 df-mi 7501 df-lti 7502 df-mpq 7540 df-enq 7542 df-nqqs 7543 df-mqqs 7545 df-1nqqs 7546 df-rq 7547 df-ltnqqs 7548 df-inp 7661 df-imp 7664

This theorem is referenced by: mullocprlem 7765 mulclpr 7767

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