addnqprl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > addnqprl		GIF version

Theorem addnqprl 7470

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 7416	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		addnqprllem 7468	. . . . . 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 400	. . . . 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 469	. . . 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 7416	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
6		addnqprllem 7468	. . . . . 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 400	. . . . 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 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 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 7409	. . . . 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 7316	. . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +_Q 𝑠) ∈ Q)
16	14, 15	genpprecll 7455	. . . 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 7422	. . . . . . . . 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 480	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
23		elprnql 7422	. . . . . . . . 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 481	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
26		addclnq 7316	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
27	22, 25, 26	syl2anc 409	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 +_Q 𝐻) ∈ Q)
28		recclnq 7333	. . . . . 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 7325	. . . . 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 1228	. . . 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 7317	. . . . . 6 ⊢ ((𝑋 ∈ Q ∧ (_Q‘(𝐺 +_Q 𝐻)) ∈ Q) → (𝑋 ·_Q (_Q‘(𝐺 +_Q 𝐻))) ∈ Q)
33	19, 29, 32	syl2anc 409	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (1^st ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·_Q (*_Q‘(𝐺 +_Q 𝐻))) ∈ Q)
34		distrnqg 7328	. . . . 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 1228	. . . 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 7324	. . . . . . . 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 409	. . . . . . 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 7334	. . . . . . . 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 2198	. . . . . 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 5858	. . . . 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 7330	. . . . . 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 2198	. . . 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 2206	. . 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 2235	. 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 1343 ∈ wcel 2136 ⟨cop 3579 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 1_Qc1q 7222 +_Q cplq 7223 ·_Q cmq 7224 *_Qcrq 7225 <_Q cltq 7226 Pcnp 7232 +_P cpp 7234

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-eprel 4267 df-id 4271 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-1o 6384 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-lti 7248 df-plpq 7285 df-mpq 7286 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-mqqs 7291 df-1nqqs 7292 df-rq 7293 df-ltnqqs 7294 df-inp 7407 df-iplp 7409

This theorem is referenced by: addlocprlemlt 7472 addclpr 7478

W3C validator