Theorem distrlem5pru 7605

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

Assertion

Ref	Expression
distrlem5pru	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Proof of Theorem distrlem5pru

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 7590	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
2	1	3adant3 1019	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
3		mulclpr 7590	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
4	3	3adant2 1018	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
5		df-iplp 7486	. . . . 5 ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑥) ∧ ℎ ∈ (1st ‘𝑦) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑥) ∧ ℎ ∈ (2nd ‘𝑦) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
6		addclnq 7393	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
7	5, 6	genpelvu 7531	. . . 4 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
8	2, 4, 7	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
9		df-imp 7487	. . . . . . . 8 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑤) ∧ ℎ ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑔 ·Q ℎ))}, {𝑥 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑤) ∧ ℎ ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑔 ·Q ℎ))}⟩)
10		mulclnq 7394	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
11	9, 10	genpelvu 7531	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12	11	3adant2 1018	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧)))
13	12	anbi2d 464	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))) ↔ (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧))))
14		df-imp 7487	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑤) ∧ ℎ ∈ (1st ‘𝑣) ∧ 𝑓 = (𝑔 ·Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑤) ∧ ℎ ∈ (2nd ‘𝑣) ∧ 𝑓 = (𝑔 ·Q ℎ))}⟩)
15	14, 10	genpelvu 7531	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16	15	3adant3 1019	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17		distrlem4pru 7603	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
18		oveq12 5900	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
19	18	eqeq2d 2201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20		eleq1 2252	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
21	19, 20	biimtrdi 163	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
22	21	imp 124	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
23	17, 22	syl5ibrcom 157	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
24	23	exp4b 367	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
25	24	com3l 81	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
26	25	exp4b 367	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) → ((𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
27	26	com23 78	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
28	27	rexlimivv 2613	. . . . . . . . 9 ⊢ (∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))))
29	28	rexlimdvv 2614	. . . . . . . 8 ⊢ (∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
30	29	com3r 79	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
31	16, 30	sylbid 150	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) → (∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
32	31	impd 254	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
33	13, 32	sylbid 150	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
34	33	rexlimdvv 2614	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
35	8, 34	sylbid 150	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
36	35	ssrdv 3176	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   ⊆ wss 3144  ‘cfv 5231  (class class class)co 5891  2^nd c2nd 6158   +_Q cplq 7300   ·_Q cmq 7301  Pcnp 7309   +_P cpp 7311   ·_P cmp 7312

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7322  df-pli 7323  df-mi 7324  df-lti 7325  df-plpq 7362  df-mpq 7363  df-enq 7365  df-nqqs 7366  df-plqqs 7367  df-mqqs 7368  df-1nqqs 7369  df-rq 7370  df-ltnqqs 7371  df-enq0 7442  df-nq0 7443  df-0nq0 7444  df-plq0 7445  df-mq0 7446  df-inp 7484  df-iplp 7486  df-imp 7487

This theorem is referenced by:  distrprg  7606