Theorem genpv 11039

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 7438	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2		rexeq 3322	. . . . 5 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
3	2	abbidv 2808	. . . 4 ⊢ (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
4	1, 3	eqeq12d 2753	. . 3 ⊢ (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)}))
5		oveq2 7439	. . . 4 ⊢ (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6		rexeq 3322	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
7	6	rexbidv 3179	. . . . 5 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
8	7	abbidv 2808	. . . 4 ⊢ (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
9	5, 8	eqeq12d 2753	. . 3 ⊢ (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)}))
10		elprnq 11031	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) → 𝑦 ∈ Q)
11		elprnq 11031	. . . . . . . . 9 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ 𝑔) → 𝑧 ∈ Q)
12		genp.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
13		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦𝐺𝑧) → (𝑥 ∈ Q ↔ (𝑦𝐺𝑧) ∈ Q))
14	12, 13	syl5ibrcom 247	. . . . . . . . 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 660	. . . . . . 7 ⊢ (((𝑓 ∈ P ∧ 𝑔 ∈ P) ∧ (𝑦 ∈ 𝑓 ∧ 𝑧 ∈ 𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
17	16	rexlimdvva 3213	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
18	17	abssdv 4068	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19		nqex 10963	. . . . 5 ⊢ Q ∈ V
20		ssexg 5323	. . . . 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 3322	. . . . . 6 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)))
23	22	abbidv 2808	. . . . 5 ⊢ (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
24		rexeq 3322	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
25	24	rexbidv 3179	. . . . . 6 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
26	25	abbidv 2808	. . . . 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 7592	. . . 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 1464	. . 3 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
30	4, 9, 29	vtocl2ga 3578	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
31		eqeq1 2741	. . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32	31	2rexbidv 3222	. . . 4 ⊢ (𝑥 = 𝑓 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑓 = (𝑦𝐺𝑧)))
33		oveq1 7438	. . . . . 6 ⊢ (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
34	33	eqeq2d 2748	. . . . 5 ⊢ (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35		oveq2 7439	. . . . . 6 ⊢ (𝑧 = ℎ → (𝑔𝐺𝑧) = (𝑔𝐺ℎ))
36	35	eqeq2d 2748	. . . . 5 ⊢ (𝑧 = ℎ → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺ℎ)))
37	34, 36	cbvrex2vw 3242	. . . 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 2812	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}
40	30, 39	eqtrdi 2793	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  (class class class)co 7431   ∈ cmpo 7433  Qcnq 10892  Pcnp 10899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-ni 10912  df-nq 10952  df-np 11021

This theorem is referenced by:  genpelv  11040  plpv  11050  mpv  11051