Theorem addnqpru 7897

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 7842	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		addnqprulem 7895	. . . . . 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 406	. . . . 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 481	. . . 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 7842	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
6		addnqprulem 7895	. . . . . 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 406	. . . . 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 480	. . . 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 7835	. . . . 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 7742	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +Q 𝑠) ∈ Q)
16	14, 15	genppreclu 7882	. . . 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 7849	. . . . . . . . 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 492	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
23		elprnqu 7849	. . . . . . . . 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 493	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
26		addclnq 7742	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 +Q 𝐻) ∈ Q)
27	22, 25, 26	syl2anc 415	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 +Q 𝐻) ∈ Q)
28		recclnq 7759	. . . . . 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 7751	. . . . 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 1278	. . . 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 7743	. . . . . 6 ⊢ ((𝑋 ∈ Q ∧ (*Q‘(𝐺 +Q 𝐻)) ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
33	19, 29, 32	syl2anc 415	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q (*Q‘(𝐺 +Q 𝐻))) ∈ Q)
34		distrnqg 7754	. . . . 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 1278	. . . 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 7750	. . . . . . . 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 415	. . . . . . 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 7760	. . . . . . . 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 2271	. . . . . 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 6101	. . . . 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 7756	. . . . . 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 2271	. . . 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 2279	. . 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 2307	. 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))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ⟨cop 3712   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7647  1_Qc1q 7648   +_Q cplq 7649   ·_Q cmq 7650  *_Qcrq 7651   <_Q cltq 7652  Pcnp 7658   +_P cpp 7660

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-inp 7833  df-iplp 7835

This theorem is used by:  addlocprlemeq  7900  addlocprlemgt  7901  addclpr  7904