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Theorem distrlem5prl 7729

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 7715	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
2	1	3adant3 1020	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
3		mulclpr 7715	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
4	3	3adant2 1019	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
5		df-iplp 7611	. . . . 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 7518	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
7	5, 6	genpelvl 7655	. . . 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 7612	. . . . . . . 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 7519	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
11	9, 10	genpelvl 7655	. . . . . . 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 1019	. . . . . 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 7612	. . . . . . . . 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 7655	. . . . . . . 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 1020	. . . . . . 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 7727	. . . . . . . . . . . . . . 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 5971	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑣 +_Q 𝑢) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
19	18	eqeq2d 2218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
20		eleq1 2269	. . . . . . . . . . . . . . . . 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 2630	. . . . . . . . 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 2631	. . . . . . . 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 2631	. . 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 3203	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 981 = wceq 1373 ∈ wcel 2177 ∃wrex 2486 ⊆ wss 3170 ‘cfv 5285 (class class class)co 5962 1^st c1st 6242 +_Q cplq 7425 ·_Q cmq 7426 Pcnp 7434 +_P cpp 7436 ·_P cmp 7437

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-nul 4181 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-iinf 4649

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-int 3895 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-tr 4154 df-eprel 4349 df-id 4353 df-po 4356 df-iso 4357 df-iord 4426 df-on 4428 df-suc 4431 df-iom 4652 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-ov 5965 df-oprab 5966 df-mpo 5967 df-1st 6244 df-2nd 6245 df-recs 6409 df-irdg 6474 df-1o 6520 df-2o 6521 df-oadd 6524 df-omul 6525 df-er 6638 df-ec 6640 df-qs 6644 df-ni 7447 df-pli 7448 df-mi 7449 df-lti 7450 df-plpq 7487 df-mpq 7488 df-enq 7490 df-nqqs 7491 df-plqqs 7492 df-mqqs 7493 df-1nqqs 7494 df-rq 7495 df-ltnqqs 7496 df-enq0 7567 df-nq0 7568 df-0nq0 7569 df-plq0 7570 df-mq0 7571 df-inp 7609 df-iplp 7611 df-imp 7612

This theorem is referenced by: distrprg 7731

W3C validator