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Theorem distrlem5prl 7805

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 7791	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
2	1	3adant3 1043	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
3		mulclpr 7791	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
4	3	3adant2 1042	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
5		df-iplp 7687	. . . . 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 7594	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
7	5, 6	genpelvl 7731	. . . 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 7688	. . . . . . . 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 7595	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
11	9, 10	genpelvl 7731	. . . . . . 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 1042	. . . . . 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 7688	. . . . . . . . 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 7731	. . . . . . . 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 1043	. . . . . . 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 7803	. . . . . . . . . . . . . . 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 6026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑣 +_Q 𝑢) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
19	18	eqeq2d 2243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
20		eleq1 2294	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . 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 2657	. . . . . . . 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 2657	. . 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 3233	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 1004 = wceq 1397 ∈ wcel 2202 ∃wrex 2511 ⊆ wss 3200 ‘cfv 5326 (class class class)co 6017 1^st c1st 6300 +_Q cplq 7501 ·_Q cmq 7502 Pcnp 7510 +_P cpp 7512 ·_P cmp 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-po 4393 df-iso 4394 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 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-1o 6581 df-2o 6582 df-oadd 6585 df-omul 6586 df-er 6701 df-ec 6703 df-qs 6707 df-ni 7523 df-pli 7524 df-mi 7525 df-lti 7526 df-plpq 7563 df-mpq 7564 df-enq 7566 df-nqqs 7567 df-plqqs 7568 df-mqqs 7569 df-1nqqs 7570 df-rq 7571 df-ltnqqs 7572 df-enq0 7643 df-nq0 7644 df-0nq0 7645 df-plq0 7646 df-mq0 7647 df-inp 7685 df-iplp 7687 df-imp 7688

This theorem is referenced by: distrprg 7807

W3C validator