Theorem distrlem5pru 7395

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 7380	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
2	1	3adant3 1001	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
3		mulclpr 7380	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
4	3	3adant2 1000	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
5		df-iplp 7276	. . . . 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 7183	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
7	5, 6	genpelvu 7321	. . . 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 408	. . 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 7277	. . . . . . . 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 7184	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
11	9, 10	genpelvu 7321	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd ‘𝐴)∃𝑧 ∈ (2nd ‘𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12	11	3adant2 1000	. . . . . 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 459	. . . . 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 7277	. . . . . . . . 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 7321	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16	15	3adant3 1001	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑦 ∈ (2nd ‘𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17		distrlem4pru 7393	. . . . . . . . . . . . . . 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 5783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
19	18	eqeq2d 2151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20		eleq1 2202	. . . . . . . . . . . . . . . . 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 162	. . . . . . . . . . . . . . . 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 123	. . . . . . . . . . . . . . 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 156	. . . . . . . . . . . . . 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 364	. . . . . . . . . . . . 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 364	. . . . . . . . . . 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 2555	. . . . . . . . 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 2556	. . . . . . . 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 149	. . . . . 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 252	. . . . 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 149	. . . 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 2556	. . 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 149	. 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 3103	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 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∃wrex 2417   ⊆ wss 3071  ‘cfv 5123  (class class class)co 5774  2^nd c2nd 6037   +_Q cplq 7090   ·_Q cmq 7091  Pcnp 7099   +_P cpp 7101   ·_P cmp 7102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-imp 7277

This theorem is referenced by:  distrprg  7396