distrlem5prl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > distrlem5prl		GIF version

Theorem distrlem5prl 7953

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 7939	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
2	1	3adant3 1048	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
3		mulclpr 7939	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
4	3	3adant2 1047	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
5		df-iplp 7835	. . . . 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 7742	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
7	5, 6	genpelvl 7879	. . . 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 415	. . 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 7836	. . . . . . . 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 7743	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
11	9, 10	genpelvl 7879	. . . . . . 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 1047	. . . . . 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 468	. . . . 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 7836	. . . . . . . . 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 7879	. . . . . . . 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 1048	. . . . . . 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 7951	. . . . . . . . . . . . . . 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 6094	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑣 +_Q 𝑢) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
19	18	eqeq2d 2250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·_Q 𝑦) ∧ 𝑢 = (𝑓 ·_Q 𝑧)) → (𝑤 = (𝑣 +_Q 𝑢) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
20		eleq1 2301	. . . . . . . . . . . . . . . . 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 2674	. . . . . . . . 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 2675	. . . . . . . 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 2675	. . 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 3254	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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 ⊆ wss 3220 ‘cfv 5377 (class class class)co 6085 1^st c1st 6372 +_Q cplq 7649 ·_Q cmq 7650 Pcnp 7658 +_P cpp 7660 ·_P cmp 7661

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-eprel 4434 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-1o 6687 df-2o 6688 df-oadd 6691 df-omul 6692 df-er 6807 df-ec 6809 df-qs 6813 df-ni 7671 df-pli 7672 df-mi 7673 df-lti 7674 df-plpq 7711 df-mpq 7712 df-enq 7714 df-nqqs 7715 df-plqqs 7716 df-mqqs 7717 df-1nqqs 7718 df-rq 7719 df-ltnqqs 7720 df-enq0 7791 df-nq0 7792 df-0nq0 7793 df-plq0 7794 df-mq0 7795 df-inp 7833 df-iplp 7835 df-imp 7836

This theorem is used by: distrprg 7955

W3C validator