Theorem distrlem5pr 11139

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 11132	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
2	1	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
3		mulclpr 11132	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
4		df-plp 11095	. . . . 5 ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑥 ∃ℎ ∈ 𝑦 𝑓 = (𝑔 +Q ℎ)})
5		addclnq 11057	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
6	4, 5	genpelv 11112	. . . 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 1364	. . 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 11096	. . . . . . . 8 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑣 𝑥 = (𝑔 ·Q ℎ)})
9		mulclnq 11059	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
10	8, 9	genpelv 11112	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧)))
11	10	3adant2 1149	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝑢 = (𝑓 ·Q 𝑧)))
12	11	anbi2d 642	. . . . 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 11096	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑣 𝑓 = (𝑔 ·Q ℎ)})
14	13, 9	genpelv 11112	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦)))
15	14	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16		distrlem4pr 11138	. . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
18	17	eqeq2d 2774	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
19		eleq1 2852	. . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 250	. . . . . . . . . . . . . 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 436	. . . . . . . . . . . . 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 90	. . . . . . . . . . . 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 436	. . . . . . . . . . 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 87	. . . . . . . . . 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 3208	. . . . . . . . 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 3222	. . . . . . . 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 88	. . . . . . 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 243	. . . . . 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 416	. . . . 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 243	. . . 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 3222	. . 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 243	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
35	34	ssrdv 3919	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  (class class class)co 7443   +_Q cplq 10967   ·_Q cmq 10968  Pcnp 10971   +_P cpp 10973   ·_P cmp 10974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8745  df-ni 10984  df-pli 10985  df-mi 10986  df-lti 10987  df-plpq 11020  df-mpq 11021  df-ltpq 11022  df-enq 11023  df-nq 11024  df-erq 11025  df-plq 11026  df-mq 11027  df-1nq 11028  df-rq 11029  df-ltnq 11030  df-np 11093  df-plp 11095  df-mp 11096

This theorem is used by:  distrpr  11140