Theorem addnqpru 7752

Description: Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)

Assertion

Ref	Expression
addnqpru	⊢ ((((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 addnqpru

Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7697	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		addnqprulem 7750	. . . . . 6 ⊢ (((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴)))
3	1, 2	sylanl1 402	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴)))
4	3	adantlr 477	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴)))
5		prop 7697	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		addnqprulem 7750	. . . . . 6 ⊢ (((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵)))
7	5, 6	sylanl1 402	. . . . 5 ⊢ (((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵)))
8	7	adantll 476	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵)))
9	4, 8	jcad 307	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵))))
10		simpl 109	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))))
11		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐴 ∈ P)
12		simpl 109	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐵 ∈ P)
13	11, 12	anim12i 338	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
14		df-iplp 7690	. . . . 5 ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1st ‘𝑥) ∧ 𝑠 ∈ (1st ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2nd ‘𝑥) ∧ 𝑠 ∈ (2nd ‘𝑦) ∧ 𝑞 = (𝑟 +Q 𝑠))}⟩)
15		addclnq 7597	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +Q 𝑠) ∈ Q)
16	14, 15	genppreclu 7737	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
17	10, 13, 16	3syl 17	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) ∈ (2nd ‘𝐴) ∧ ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻) ∈ (2nd ‘𝐵)) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
18	9, 17	syld 45	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) <Q 𝑋 → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵))))
19		simpr 110	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
20		elprnqu 7704	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
21	1, 20	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
22	21	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
23		elprnqu 7704	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
24	5, 23	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
25	24	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
26		addclnq 7597	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 +Q 𝐻) ∈ Q)
27	22, 25, 26	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 +Q 𝐻) ∈ Q)
28		recclnq 7614	. . . . . 6 ⊢ ((𝐺 +Q 𝐻) ∈ Q → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
29	27, 28	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (*Q‘(𝐺 +Q 𝐻)) ∈ Q)
30		mulassnqg 7606	. . . . 5 ⊢ ((𝑋 ∈ Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
31	19, 29, 27, 30	syl3anc 1273	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))))
32		mulclnq 7598	. . . . . 6 ⊢ ((𝑋 ∈ Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
33	19, 29, 32	syl2anc 411	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
34		distrnqg 7609	. . . . 5 ⊢ (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q ∧ 𝐺 ∈ Q ∧ 𝐻 ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
35	33, 22, 25, 34	syl3anc 1273	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q (𝐺 +Q 𝐻)) = (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)))
36		mulcomnqg 7605	. . . . . . . 8 ⊢ (((*Q‘(𝐺 +Q 𝐻)) ∈ Q ∧ (𝐺 +Q 𝐻) ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
37	29, 27, 36	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))))
38		recidnq 7615	. . . . . . . 8 ⊢ ((𝐺 +Q 𝐻) ∈ Q → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
39	27, 38	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +Q 𝐻) ·Q (*Q‘(𝐺 +Q 𝐻))) = 1Q)
40	37, 39	eqtrd 2263	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻)) = 1Q)
41	40	oveq2d 6036	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = (𝑋 ·Q 1Q))
42		mulidnq 7611	. . . . . 6 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
43	42	adantl 277	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
44	41, 43	eqtrd 2263	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘(𝐺 +Q 𝐻)) ·Q (𝐺 +Q 𝐻))) = 𝑋)
45	31, 35, 44	3eqtr3d 2271	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) = 𝑋)
46	45	eleq1d 2299	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐺) +Q ((𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ·Q 𝐻)) ∈ (2nd ‘(𝐴 +P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 +P 𝐵))))
47	18, 46	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   = wceq 1397   ∈ wcel 2201  ⟨cop 3671   class class class wbr 4087  ‘cfv 5325  (class class class)co 6020  1^st c1st 6303  2^nd c2nd 6304  Qcnq 7502  1_Qc1q 7503   +_Q cplq 7504   ·_Q cmq 7505  *_Qcrq 7506   <_Q cltq 7507  Pcnp 7513   +_P cpp 7515

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-eprel 4385  df-id 4389  df-iord 4462  df-on 4464  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-irdg 6538  df-1o 6584  df-oadd 6588  df-omul 6589  df-er 6704  df-ec 6706  df-qs 6710  df-ni 7526  df-pli 7527  df-mi 7528  df-lti 7529  df-plpq 7566  df-mpq 7567  df-enq 7569  df-nqqs 7570  df-plqqs 7571  df-mqqs 7572  df-1nqqs 7573  df-rq 7574  df-ltnqqs 7575  df-inp 7688  df-iplp 7690

This theorem is referenced by:  addlocprlemeq  7755  addlocprlemgt  7756  addclpr  7759