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Theorem distrlem5prl 7769

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

**Assertion**
Ref	Expression
distrlem5prl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))) ⊆ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))

Proof of Theorem distrlem5prl Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulclpr 7755	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
2	1	3adant3 1041	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
3		mulclpr 7755	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
4	3	3adant2 1040	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
5		df-iplp 7651	. . . . 5 ⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑥) ∧ ℎ ∈ (1^st ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑥) ∧ ℎ ∈ (2^nd ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}⟩)
6		addclnq 7558	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
7	5, 6	genpelvl 7695	. . . 4 ⊢ (((𝐴 ·_P 𝐵) ∈ P ∧ (𝐴 ·_P 𝐶) ∈ P) → (𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))) ↔ ∃𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵))∃𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶))𝑤 = (𝑣 +_Q 𝑢)))
8	2, 4, 7	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))) ↔ ∃𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵))∃𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶))𝑤 = (𝑣 +_Q 𝑢)))
9		df-imp 7652	. . . . . . . 8 ⊢ ·_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑤) ∧ ℎ ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑔 ·_Q ℎ))}, {𝑥 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑤) ∧ ℎ ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑔 ·_Q ℎ))}⟩)
10		mulclnq 7559	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
11	9, 10	genpelvl 7695	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧)))
12	11	3adant2 1040	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶)) ↔ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧)))
13	12	anbi2d 464	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶))) ↔ (𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧))))
14		df-imp 7652	. . . . . . . . 9 ⊢ ·_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑤) ∧ ℎ ∈ (1^st ‘𝑣) ∧ 𝑓 = (𝑔 ·_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑤) ∧ ℎ ∈ (2^nd ‘𝑣) ∧ 𝑓 = (𝑔 ·_Q ℎ))}⟩)
15	14, 10	genpelvl 7695	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑥 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑣 = (𝑥 ·_Q 𝑦)))
16	15	3adant3 1041	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑥 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑣 = (𝑥 ·_Q 𝑦)))
17		distrlem4prl 7767	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
18		oveq12 6009	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑣 +_Q 𝑢) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
19	18	eqeq2d 2241	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
20		eleq1 2292	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
21	19, 20	biimtrdi 163	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))))
22	21	imp 124	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) ∧ 𝑤 = (𝑣 +_Q 𝑢)) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
23	17, 22	syl5ibrcom 157	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) ∧ 𝑤 = (𝑣 +_Q 𝑢)) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
24	23	exp4b 367	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))
25	24	com3l 81	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))
26	25	exp4b 367	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → ((𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑣 = (𝑥 ·_Q 𝑦) → (𝑢 = (𝑓 ·_Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))))
27	26	com23 78	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑣 = (𝑥 ·_Q 𝑦) → ((𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑢 = (𝑓 ·_Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))))
28	27	rexlimivv 2654	. . . . . . . . 9 ⊢ (∃𝑥 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑣 = (𝑥 ·_Q 𝑦) → ((𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑢 = (𝑓 ·_Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))))))
29	28	rexlimdvv 2655	. . . . . . . 8 ⊢ (∃𝑥 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑣 = (𝑥 ·_Q 𝑦) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))
30	29	com3r 79	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ (1^st ‘𝐴)∃𝑦 ∈ (1^st ‘𝐵)𝑣 = (𝑥 ·_Q 𝑦) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))
31	16, 30	sylbid 150	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) → (∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))))
32	31	impd 254	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑓 ∈ (1^st ‘𝐴)∃𝑧 ∈ (1^st ‘𝐶)𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))))
33	13, 32	sylbid 150	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶))) → (𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))))
34	33	rexlimdvv 2655	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑣 ∈ (1^st ‘(𝐴 ·_P 𝐵))∃𝑢 ∈ (1^st ‘(𝐴 ·_P 𝐶))𝑤 = (𝑣 +_Q 𝑢) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
35	8, 34	sylbid 150	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))) → 𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
36	35	ssrdv 3230	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))) ⊆ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∃wrex 2509 ⊆ wss 3197 ‘cfv 5317 (class class class)co 6000 1^st c1st 6282 +_Q cplq 7465 ·_Q cmq 7466 Pcnp 7474 +_P cpp 7476 ·_P cmp 7477

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-eprel 4379 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-1o 6560 df-2o 6561 df-oadd 6564 df-omul 6565 df-er 6678 df-ec 6680 df-qs 6684 df-ni 7487 df-pli 7488 df-mi 7489 df-lti 7490 df-plpq 7527 df-mpq 7528 df-enq 7530 df-nqqs 7531 df-plqqs 7532 df-mqqs 7533 df-1nqqs 7534 df-rq 7535 df-ltnqqs 7536 df-enq0 7607 df-nq0 7608 df-0nq0 7609 df-plq0 7610 df-mq0 7611 df-inp 7649 df-iplp 7651 df-imp 7652

This theorem is referenced by: distrprg 7771

W3C validator