Theorem mulnqpru 7702

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

Assertion

Ref	Expression
mulnqpru	⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mulnqpru

Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 7534	. . . . . . 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 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3		prop 7608	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elprnqu 7615	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
5	3, 4	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
6	5	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
7		prop 7608	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
8		elprnqu 7615	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
9	7, 8	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
10	9	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
11		mulclnq 7509	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
12	6, 10, 11	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
13		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
14		recclnq 7525	. . . . . . 7 ⊢ (𝐻 ∈ Q → (*Q‘𝐻) ∈ Q)
15	10, 14	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (*Q‘𝐻) ∈ Q)
16		mulcomnqg 7516	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
17	16	adantl 277	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
18	2, 12, 13, 15, 17	caovord2d 6129	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻))))
19		mulassnqg 7517	. . . . . . . 8 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q ∧ (*Q‘𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
20	6, 10, 15, 19	syl3anc 1250	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
21		recidnq 7526	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
22	21	oveq2d 5973	. . . . . . . 8 ⊢ (𝐻 ∈ Q → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
23	10, 22	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
24		mulidnq 7522	. . . . . . . 8 ⊢ (𝐺 ∈ Q → (𝐺 ·Q 1Q) = 𝐺)
25	6, 24	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 1Q) = 𝐺)
26	20, 23, 25	3eqtrd 2243	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = 𝐺)
27	26	breq1d 4061	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻)) ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
28	18, 27	bitrd 188	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
29		prcunqu 7618	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
30	3, 29	sylan 283	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
31	30	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
32	28, 31	sylbid 150	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
33		df-imp 7602	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}⟩)
34		mulclnq 7509	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
35	33, 34	genppreclu 7648	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) ∧ 𝐻 ∈ (2nd ‘𝐵)) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
36	35	exp4b 367	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → (𝐻 ∈ (2nd ‘𝐵) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
37	36	com34 83	. . . . . 6 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐻 ∈ (2nd ‘𝐵) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
38	37	imp32 257	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
39	38	adantlr 477	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
40	39	adantr 276	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
41	32, 40	syld 45	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
42		mulassnqg 7517	. . . . 5 ⊢ ((𝑋 ∈ Q ∧ (*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
43	13, 15, 10, 42	syl3anc 1250	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
44		mulcomnqg 7516	. . . . . . 7 ⊢ (((*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
45	15, 10, 44	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
46	10, 21	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
47	45, 46	eqtrd 2239	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = 1Q)
48	47	oveq2d 5973	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49		mulidnq 7522	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
50	49	adantl 277	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
51	43, 48, 50	3eqtrd 2243	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = 𝑋)
52	51	eleq1d 2275	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
53	41, 52	sylibd 149	1 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ⟨cop 3641   class class class wbr 4051  ‘cfv 5280  (class class class)co 5957  1^st c1st 6237  2^nd c2nd 6238  Qcnq 7413  1_Qc1q 7414   ·_Q cmq 7416  *_Qcrq 7417   <_Q cltq 7418  Pcnp 7424   ·_P cmp 7427

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-mi 7439  df-lti 7440  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486  df-inp 7599  df-imp 7602

This theorem is referenced by:  mullocprlem  7703  mulclpr  7705