Theorem genpv 10755

Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
genp.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)

Assertion

Ref	Expression
genpv	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})

Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,ℎ,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,ℎ   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑓,𝑔,ℎ   𝑓,𝐹,𝑔

Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,ℎ)

Proof of Theorem genpv

Step	Hyp	Ref	Expression
1		oveq1 7282	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2		rexeq 3343	. . . . 5 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
3	2	abbidv 2807	. . . 4 ⊢ (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
4	1, 3	eqeq12d 2754	. . 3 ⊢ (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)}))
5		oveq2 7283	. . . 4 ⊢ (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6		rexeq 3343	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
7	6	rexbidv 3226	. . . . 5 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
8	7	abbidv 2807	. . . 4 ⊢ (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
9	5, 8	eqeq12d 2754	. . 3 ⊢ (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)}))
10		elprnq 10747	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) → 𝑦 ∈ Q)
11		elprnq 10747	. . . . . . . . 9 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ 𝑔) → 𝑧 ∈ Q)
12		genp.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
13		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦𝐺𝑧) → (𝑥 ∈ Q ↔ (𝑦𝐺𝑧) ∈ Q))
14	12, 13	syl5ibrcom 246	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
15	10, 11, 14	syl2an 596	. . . . . . . 8 ⊢ (((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) ∧ (𝑔 ∈ P ∧ 𝑧 ∈ 𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
16	15	an4s 657	. . . . . . 7 ⊢ (((𝑓 ∈ P ∧ 𝑔 ∈ P) ∧ (𝑦 ∈ 𝑓 ∧ 𝑧 ∈ 𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
17	16	rexlimdvva 3223	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
18	17	abssdv 4002	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19		nqex 10679	. . . . 5 ⊢ Q ∈ V
20		ssexg 5247	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q ∧ Q ∈ V) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
21	18, 19, 20	sylancl 586	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22		rexeq 3343	. . . . . 6 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)))
23	22	abbidv 2807	. . . . 5 ⊢ (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
24		rexeq 3343	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
25	24	rexbidv 3226	. . . . . 6 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
26	25	abbidv 2807	. . . . 5 ⊢ (𝑣 = 𝑔 → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
27		genp.1	. . . . 5 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
28	23, 26, 27	ovmpog 7432	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
29	21, 28	mpd3an3 1461	. . 3 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
30	4, 9, 29	vtocl2ga 3514	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
31		eqeq1 2742	. . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32	31	2rexbidv 3229	. . . 4 ⊢ (𝑥 = 𝑓 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑓 = (𝑦𝐺𝑧)))
33		oveq1 7282	. . . . . 6 ⊢ (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
34	33	eqeq2d 2749	. . . . 5 ⊢ (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35		oveq2 7283	. . . . . 6 ⊢ (𝑧 = ℎ → (𝑔𝐺𝑧) = (𝑔𝐺ℎ))
36	35	eqeq2d 2749	. . . . 5 ⊢ (𝑧 = ℎ → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺ℎ)))
37	34, 36	cbvrex2vw 3397	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑓 = (𝑦𝐺𝑧) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ))
38	32, 37	bitrdi 287	. . 3 ⊢ (𝑥 = 𝑓 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
39	38	cbvabv 2811	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}
40	30, 39	eqtrdi 2794	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  (class class class)co 7275   ∈ cmpo 7277  Qcnq 10608  Pcnp 10615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-ni 10628  df-nq 10668  df-np 10737

This theorem is referenced by:  genpelv  10756  plpv  10766  mpv  10767