Theorem genpv 10887

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 7353	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2		rexeq 3288	. . . . 5 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
3	2	abbidv 2797	. . . 4 ⊢ (𝑓 = 𝐴 → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
4	1, 3	eqeq12d 2747	. . 3 ⊢ (𝑓 = 𝐴 → ((𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)}))
5		oveq2 7354	. . . 4 ⊢ (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6		rexeq 3288	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
7	6	rexbidv 3156	. . . . 5 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
8	7	abbidv 2797	. . . 4 ⊢ (𝑔 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
9	5, 8	eqeq12d 2747	. . 3 ⊢ (𝑔 = 𝐵 → ((𝐴𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ↔ (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)}))
10		elprnq 10879	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) → 𝑦 ∈ Q)
11		elprnq 10879	. . . . . . . . 9 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ 𝑔) → 𝑧 ∈ Q)
12		genp.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
13		eleq1 2819	. . . . . . . . . 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 3189	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
18	17	abssdv 4019	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19		nqex 10811	. . . . 5 ⊢ Q ∈ V
20		ssexg 5261	. . . . 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 3288	. . . . . 6 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)))
23	22	abbidv 2797	. . . . 5 ⊢ (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
24		rexeq 3288	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
25	24	rexbidv 3156	. . . . . 6 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
26	25	abbidv 2797	. . . . 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 7505	. . . 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 3533	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)})
31		eqeq1 2735	. . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑦𝐺𝑧)))
32	31	2rexbidv 3197	. . . 4 ⊢ (𝑥 = 𝑓 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑓 = (𝑦𝐺𝑧)))
33		oveq1 7353	. . . . . 6 ⊢ (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
34	33	eqeq2d 2742	. . . . 5 ⊢ (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35		oveq2 7354	. . . . . 6 ⊢ (𝑧 = ℎ → (𝑔𝐺𝑧) = (𝑔𝐺ℎ))
36	35	eqeq2d 2742	. . . . 5 ⊢ (𝑧 = ℎ → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺ℎ)))
37	34, 36	cbvrex2vw 3215	. . . 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 2801	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}
40	30, 39	eqtrdi 2782	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  Vcvv 3436   ⊆ wss 3902  (class class class)co 7346   ∈ cmpo 7348  Qcnq 10740  Pcnp 10747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-ni 10760  df-nq 10800  df-np 10869

This theorem is referenced by:  genpelv  10888  plpv  10898  mpv  10899