Theorem mulnqpru 6810

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 6642	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
2	1	adantl 271	. . . . . 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 6716	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elprnqu 6723	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
5	3, 4	sylan 277	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
6	5	ad2antrr 472	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
7		prop 6716	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
8		elprnqu 6723	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
9	7, 8	sylan 277	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
10	9	ad2antlr 473	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
11		mulclnq 6617	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
12	6, 10, 11	syl2anc 403	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
13		simpr 108	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
14		recclnq 6633	. . . . . . 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 6624	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
17	16	adantl 271	. . . . . 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 5695	. . . . 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 6625	. . . . . . . 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 1170	. . . . . . 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 6634	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
22	21	oveq2d 5553	. . . . . . . 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 6630	. . . . . . . 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 2118	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = 𝐺)
27	26	breq1d 3797	. . . . 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 186	. . . 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 6726	. . . . . 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 277	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
31	30	ad2antrr 472	. . . 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 148	. . 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 6710	. . . . . . . . 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 6617	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
35	33, 34	genppreclu 6756	. . . . . . . 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 359	. . . . . . 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 82	. . . . . 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 253	. . . . 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 461	. . . 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 270	. . 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 44	. 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 6625	. . . . 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 1170	. . . 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 6624	. . . . . . 7 ⊢ (((*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
45	15, 10, 44	syl2anc 403	. . . . . 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 2114	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = 1Q)
48	47	oveq2d 5553	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49		mulidnq 6630	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
50	49	adantl 271	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
51	43, 48, 50	3eqtrd 2118	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = 𝑋)
52	51	eleq1d 2148	. 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 147	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 102   ↔ wb 103   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ⟨cop 3403   class class class wbr 3787  ‘cfv 4926  (class class class)co 5537  1^st c1st 5790  2^nd c2nd 5791  Qcnq 6521  1_Qc1q 6522   ·_Q cmq 6524  *_Qcrq 6525   <_Q cltq 6526  Pcnp 6532   ·_P cmp 6535

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-mi 6547  df-lti 6548  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-inp 6707  df-imp 6710

This theorem is referenced by:  mullocprlem  6811  mulclpr  6813