mulnqprl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > mulnqprl		GIF version

Theorem mulnqprl 7942

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 7775	. . . . . . 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 7849	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
5		elprnql 7855	. . . . . . . . 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 492	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
8		prop 7849	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
9		elprnql 7855	. . . . . . . . 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 493	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
12		mulclnq 7750	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
13	7, 11, 12	syl2anc 415	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·_Q 𝐻) ∈ Q)
14		recclnq 7766	. . . . . . 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 7757	. . . . . . 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 6262	. . . . 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 7758	. . . . . . . 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 1278	. . . . . . 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 7767	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·_Q (*_Q‘𝐻)) = 1_Q)
22	21	oveq2d 6104	. . . . . . . 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 7763	. . . . . . . 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 2275	. . . . . 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 4143	. . . . 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 7858	. . . . . 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 492	. . . 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 7843	. . . . . . . . 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 7750	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
35	33, 34	genpprecll 7888	. . . . . . . 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 481	. . . 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 7758	. . . . 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 1278	. . . 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 7757	. . . . . . 7 ⊢ (((_Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((_Q‘𝐻) ·_Q 𝐻) = (𝐻 ·_Q (*_Q‘𝐻)))
45	15, 11, 44	syl2anc 415	. . . . . 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 2271	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘𝐻) ·_Q 𝐻) = 1_Q)
48	47	oveq2d 6104	. . . 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 7763	. . . . 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 2275	. . 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 2307	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ⟨cop 3712 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 Qcnq 7654 1_Qc1q 7655 ·_Q cmq 7657 *_Qcrq 7658 <_Q cltq 7659 Pcnp 7665 ·_P cmp 7668

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-1o 6691 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7678 df-mi 7680 df-lti 7681 df-mpq 7719 df-enq 7721 df-nqqs 7722 df-mqqs 7724 df-1nqqs 7725 df-rq 7726 df-ltnqqs 7727 df-inp 7840 df-imp 7843

This theorem is used by: mullocprlem 7944 mulclpr 7946

W3C validator