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Theorem distrlem5prl 7587

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 7573	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
2	1	3adant3 1017	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
3		mulclpr 7573	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
4	3	3adant2 1016	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
5		df-iplp 7469	. . . . 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 7376	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
7	5, 6	genpelvl 7513	. . . 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 7470	. . . . . . . 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 7377	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
11	9, 10	genpelvl 7513	. . . . . . 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 1016	. . . . . 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 7470	. . . . . . . . 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 7513	. . . . . . . 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 1017	. . . . . . 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 7585	. . . . . . . . . . . . . . 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 5886	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑣 +_Q 𝑢) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
19	18	eqeq2d 2189	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
20		eleq1 2240	. . . . . . . . . . . . . . . . 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 2600	. . . . . . . . 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 2601	. . . . . . . 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 2601	. . 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 3163	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 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3131 ‘cfv 5218 (class class class)co 5877 1^st c1st 6141 +_Q cplq 7283 ·_Q cmq 7284 Pcnp 7292 +_P cpp 7294 ·_P cmp 7295

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-2o 6420 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-mpq 7346 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-mqqs 7351 df-1nqqs 7352 df-rq 7353 df-ltnqqs 7354 df-enq0 7425 df-nq0 7426 df-0nq0 7427 df-plq0 7428 df-mq0 7429 df-inp 7467 df-iplp 7469 df-imp 7470

This theorem is referenced by: distrprg 7589

W3C validator