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Theorem addnqprl 7337

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

**Assertion**
Ref	Expression
addnqprl	⊢ ((((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 addnqprl Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7283	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		addnqprllem 7335	. . . . . 6 ⊢ (((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴)))
3	1, 2	sylanl1 399	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴)))
4	3	adantlr 468	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴)))
5		prop 7283	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
6		addnqprllem 7335	. . . . . 6 ⊢ (((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵)))
7	5, 6	sylanl1 399	. . . . 5 ⊢ (((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵)))
8	7	adantll 467	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → ((𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵)))
9	4, 8	jcad 305	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 <_Q (𝐺 +_Q 𝐻) → (((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴) ∧ ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵))))
10		simpl 108	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))))
11		simpl 108	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐴 ∈ P)
12		simpl 108	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐵 ∈ P)
13	11, 12	anim12i 336	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
14		df-iplp 7276	. . . . 5 ⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (1^st ‘𝑥) ∧ 𝑠 ∈ (1^st ‘𝑦) ∧ 𝑞 = (𝑟 +_Q 𝑠))}, {𝑞 ∈ Q ∣ ∃𝑟 ∈ Q ∃𝑠 ∈ Q (𝑟 ∈ (2^nd ‘𝑥) ∧ 𝑠 ∈ (2^nd ‘𝑦) ∧ 𝑞 = (𝑟 +_Q 𝑠))}⟩)
15		addclnq 7183	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +_Q 𝑠) ∈ Q)
16	14, 15	genpprecll 7322	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴) ∧ ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵)) → (((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) +_Q ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻)) ∈ (1^st ‘(𝐴 +_P 𝐵))))
17	10, 13, 16	3syl 17	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) ∈ (1^st ‘𝐴) ∧ ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻) ∈ (1^st ‘𝐵)) → (((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) +_Q ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻)) ∈ (1^st ‘(𝐴 +_P 𝐵))))
18	9, 17	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‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) +_Q ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻)) ∈ (1^st ‘(𝐴 +_P 𝐵))))
19		simpr 109	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
20		elprnql 7289	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
21	1, 20	sylan 281	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
22	21	ad2antrr 479	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
23		elprnql 7289	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
24	5, 23	sylan 281	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
25	24	ad2antlr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
26		addclnq 7183	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
27	22, 25, 26	syl2anc 408	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
28		recclnq 7200	. . . . . 6 ⊢ ((𝐺 +_Q 𝐻) ∈ Q → (*_Q‘(𝐺 +_Q 𝐻)) ∈ Q)
29	27, 28	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (*_Q‘(𝐺 +_Q 𝐻)) ∈ Q)
30		mulassnqg 7192	. . . . 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 1216	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘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 7184	. . . . . 6 ⊢ ((𝑋 ∈ Q ∧ (_Q‘(𝐺 +_Q 𝐻)) ∈ Q) → (𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ∈ Q)
33	19, 29, 32	syl2anc 408	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ∈ Q)
34		distrnqg 7195	. . . . 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 1216	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘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 7191	. . . . . . . 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 408	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((_Q‘(𝐺 +_Q 𝐻)) ·_Q (𝐺 +_Q 𝐻)) = ((𝐺 +_Q 𝐻) ·_Q (_Q‘(𝐺 +_Q 𝐻))))
38		recidnq 7201	. . . . . . . 8 ⊢ ((𝐺 +_Q 𝐻) ∈ Q → ((𝐺 +_Q 𝐻) ·_Q (*_Q‘(𝐺 +_Q 𝐻))) = 1_Q)
39	27, 38	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 +_Q 𝐻) ·_Q (*_Q‘(𝐺 +_Q 𝐻))) = 1_Q)
40	37, 39	eqtrd 2172	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*_Q‘(𝐺 +_Q 𝐻)) ·_Q (𝐺 +_Q 𝐻)) = 1_Q)
41	40	oveq2d 5790	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q ((*_Q‘(𝐺 +_Q 𝐻)) ·_Q (𝐺 +_Q 𝐻))) = (𝑋 ·_Q 1_Q))
42		mulidnq 7197	. . . . . 6 ⊢ (𝑋 ∈ Q → (𝑋 ·_Q 1_Q) = 𝑋)
43	42	adantl 275	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q 1_Q) = 𝑋)
44	41, 43	eqtrd 2172	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q ((*_Q‘(𝐺 +_Q 𝐻)) ·_Q (𝐺 +_Q 𝐻))) = 𝑋)
45	31, 35, 44	3eqtr3d 2180	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) +_Q ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻)) = 𝑋)
46	45	eleq1d 2208	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → ((((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐺) +_Q ((𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ·_Q 𝐻)) ∈ (1^st ‘(𝐴 +_P 𝐵)) ↔ 𝑋 ∈ (1^st ‘(𝐴 +_P 𝐵))))
47	18, 46	sylibd 148	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

Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ⟨cop 3530 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1^st c1st 6036 2^nd c2nd 6037 Qcnq 7088 1_Qc1q 7089 +_Q cplq 7090 ·_Q cmq 7091 *_Qcrq 7092 <_Q cltq 7093 Pcnp 7099 +_P cpp 7101

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-inp 7274 df-iplp 7276

This theorem is referenced by: addlocprlemlt 7339 addclpr 7345

W3C validator