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Theorem addnqprl 7749

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 7695	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		addnqprllem 7747	. . . . . 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 402	. . . . 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 477	. . . 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 7695	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
6		addnqprllem 7747	. . . . . 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 402	. . . . 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 476	. . . 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 307	. . 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 109	. . . 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 109	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐴 ∈ P)
12		simpl 109	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐵 ∈ P)
13	11, 12	anim12i 338	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
14		df-iplp 7688	. . . . 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 7595	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +_Q 𝑠) ∈ Q)
16	14, 15	genpprecll 7734	. . . 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 110	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
20		elprnql 7701	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
21	1, 20	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) → 𝐺 ∈ Q)
22	21	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
23		elprnql 7701	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
24	5, 23	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵)) → 𝐻 ∈ Q)
25	24	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
26		addclnq 7595	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
27	22, 25, 26	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
28		recclnq 7612	. . . . . 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 7604	. . . . 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 ∈ (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 7596	. . . . . 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 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ∈ Q)
34		distrnqg 7607	. . . . 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 ∈ (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 7603	. . . . . . . 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 ∈ (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 7613	. . . . . . . 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 2264	. . . . . 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 6034	. . . . 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 7609	. . . . . 6 ⊢ (𝑋 ∈ Q → (𝑋 ·_Q 1_Q) = 𝑋)
43	42	adantl 277	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q 1_Q) = 𝑋)
44	41, 43	eqtrd 2264	. . . 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 2272	. . 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 2300	. 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 149	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 104 = wceq 1397 ∈ wcel 2202 ⟨cop 3672 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 1^st c1st 6301 2^nd c2nd 6302 Qcnq 7500 1_Qc1q 7501 +_Q cplq 7502 ·_Q cmq 7503 *_Qcrq 7504 <_Q cltq 7505 Pcnp 7511 +_P cpp 7513

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-1o 6582 df-oadd 6586 df-omul 6587 df-er 6702 df-ec 6704 df-qs 6708 df-ni 7524 df-pli 7525 df-mi 7526 df-lti 7527 df-plpq 7564 df-mpq 7565 df-enq 7567 df-nqqs 7568 df-plqqs 7569 df-mqqs 7570 df-1nqqs 7571 df-rq 7572 df-ltnqqs 7573 df-inp 7686 df-iplp 7688

This theorem is referenced by: addlocprlemlt 7751 addclpr 7757

W3C validator