Theorem distrlem5pru 6685

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 6670	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
2	1	3adant3 924	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
3		mulclpr 6670	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
4	3	3adant2 923	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
5		df-iplp 6566	. . . . 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 6473	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
7	5, 6	genpelvu 6611	. . . 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 391	. . 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 6567	. . . . . . . 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 6474	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
11	9, 10	genpelvu 6611	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12	11	3adant2 923	. . . . . 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 437	. . . . 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 6567	. . . . . . . . 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 6611	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16	15	3adant3 924	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17		distrlem4pru 6683	. . . . . . . . . . . . . . 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 5521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
19	18	eqeq2d 2051	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20		eleq1 2100	. . . . . . . . . . . . . . . . 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	syl6bi 152	. . . . . . . . . . . . . . . 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 115	. . . . . . . . . . . . . . 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 146	. . . . . . . . . . . . . 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 349	. . . . . . . . . . . . 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 75	. . . . . . . . . . . 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 349	. . . . . . . . . . 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 72	. . . . . . . . . 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 2438	. . . . . . . . 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 2439	. . . . . . . 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 73	. . . . . . 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 139	. . . . . 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 242	. . . . 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 139	. . . 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 2439	. . 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 139	. 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 2951	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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307   ⊆ wss 2917  ‘cfv 4902  (class class class)co 5512  2^nd c2nd 5766   +_Q cplq 6380   ·_Q cmq 6381  Pcnp 6389   +_P cpp 6391   ·_P cmp 6392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-imp 6567

This theorem is referenced by:  distrprg  6686