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Theorem addnqprl 7613

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 7559	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		addnqprllem 7611	. . . . . 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 7559	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
6		addnqprllem 7611	. . . . . 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 7552	. . . . 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 7459	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +_Q 𝑠) ∈ Q)
16	14, 15	genpprecll 7598	. . . 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 7565	. . . . . . . . 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 7565	. . . . . . . . 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 7459	. . . . . . 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 7476	. . . . . 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 7468	. . . . 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 1249	. . . 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 7460	. . . . . 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 7471	. . . . 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 1249	. . . 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 7467	. . . . . . . 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 7477	. . . . . . . 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 2229	. . . . . 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 5941	. . . . 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 7473	. . . . . 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 2229	. . . 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 2237	. . 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 2265	. 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 1364 ∈ wcel 2167 ⟨cop 3626 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 1^st c1st 6205 2^nd c2nd 6206 Qcnq 7364 1_Qc1q 7365 +_Q cplq 7366 ·_Q cmq 7367 *_Qcrq 7368 <_Q cltq 7369 Pcnp 7375 +_P cpp 7377

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-eprel 4325 df-id 4329 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-1o 6483 df-oadd 6487 df-omul 6488 df-er 6601 df-ec 6603 df-qs 6607 df-ni 7388 df-pli 7389 df-mi 7390 df-lti 7391 df-plpq 7428 df-mpq 7429 df-enq 7431 df-nqqs 7432 df-plqqs 7433 df-mqqs 7434 df-1nqqs 7435 df-rq 7436 df-ltnqqs 7437 df-inp 7550 df-iplp 7552

This theorem is referenced by: addlocprlemlt 7615 addclpr 7621

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