Theorem distrlem5pr 10980

Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
distrlem5pr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))

Proof of Theorem distrlem5pr

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 10973	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
2	1	3adant3 1144	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
3		mulclpr 10973	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
4		df-plp 10936	. . . . 5 ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑥 ∃ℎ ∈ 𝑦 𝑓 = (𝑔 +Q ℎ)})
5		addclnq 10898	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
6	4, 5	genpelv 10953	. . . 4 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢)))
7	2, 3, 6	3imp3i2an 1358	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢)))
8		df-mp 10937	. . . . . . . 8 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑣 𝑥 = (𝑔 ·Q ℎ)})
9		mulclnq 10900	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
10	8, 9	genpelv 10953	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧)))
11	10	3adant2 1143	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧)))
12	11	anbi2d 639	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) ↔ (𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧))))
13		df-mp 10937	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑣 𝑓 = (𝑔 ·Q ℎ)})
14	13, 9	genpelv 10953	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦)))
15	14	3adant3 1144	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16		distrlem4pr 10979	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
17		oveq12 7399	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
18	17	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
19		eleq1 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
20	18, 19	biimtrdi 255	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
21	20	imp 410	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
22	16, 21	syl5ibrcom 249	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
23	22	exp4b 434	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
24	23	com3l 89	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
25	24	exp4b 434	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
26	25	com23 86	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
27	26	rexlimivv 3203	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))))
28	27	rexlimdvv 3217	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧) → ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
29	28	com3r 87	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
30	15, 29	sylbid 242	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐴 ·P 𝐵) → (∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
31	30	impd 414	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
32	12, 31	sylbid 242	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
33	32	rexlimdvv 3217	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
34	7, 33	sylbid 242	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
35	34	ssrdv 3942	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085   ⊆ wss 3904  (class class class)co 7390   +_Q cplq 10808   ·_Q cmq 10809  Pcnp 10812   +_P cpp 10814   ·_P cmp 10815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-inf2 9591

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-oadd 8434  df-omul 8435  df-er 8671  df-ni 10825  df-pli 10826  df-mi 10827  df-lti 10828  df-plpq 10861  df-mpq 10862  df-ltpq 10863  df-enq 10864  df-nq 10865  df-erq 10866  df-plq 10867  df-mq 10868  df-1nq 10869  df-rq 10870  df-ltnq 10871  df-np 10934  df-plp 10936  df-mp 10937

This theorem is referenced by:  distrpr  10981