Theorem genpv 10922

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 7375	. . . 4 ⊢ (𝑓 = 𝐴 → (𝑓𝐹𝑔) = (𝐴𝐹𝑔))
2		rexeq 3294	. . . . 5 ⊢ (𝑓 = 𝐴 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
3	2	abbidv 2803	. . . 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 7376	. . . 4 ⊢ (𝑔 = 𝐵 → (𝐴𝐹𝑔) = (𝐴𝐹𝐵))
6		rexeq 3294	. . . . . 6 ⊢ (𝑔 = 𝐵 → (∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
7	6	rexbidv 3162	. . . . 5 ⊢ (𝑔 = 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)))
8	7	abbidv 2803	. . . 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 10914	. . . . . . . . 9 ⊢ ((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) → 𝑦 ∈ Q)
11		elprnq 10914	. . . . . . . . 9 ⊢ ((𝑔 ∈ P ∧ 𝑧 ∈ 𝑔) → 𝑧 ∈ Q)
12		genp.2	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
13		eleq1 2825	. . . . . . . . . 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 597	. . . . . . . 8 ⊢ (((𝑓 ∈ P ∧ 𝑦 ∈ 𝑓) ∧ (𝑔 ∈ P ∧ 𝑧 ∈ 𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
16	15	an4s 661	. . . . . . 7 ⊢ (((𝑓 ∈ P ∧ 𝑔 ∈ P) ∧ (𝑦 ∈ 𝑓 ∧ 𝑧 ∈ 𝑔)) → (𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
17	16	rexlimdvva 3195	. . . . . 6 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧) → 𝑥 ∈ Q))
18	17	abssdv 4021	. . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q)
19		nqex 10846	. . . . 5 ⊢ Q ∈ V
20		ssexg 5270	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ⊆ Q ∧ Q ∈ V) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
21	18, 19, 20	sylancl 587	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)} ∈ V)
22		rexeq 3294	. . . . . 6 ⊢ (𝑤 = 𝑓 → (∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)))
23	22	abbidv 2803	. . . . 5 ⊢ (𝑤 = 𝑓 → {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)} = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
24		rexeq 3294	. . . . . . 7 ⊢ (𝑣 = 𝑔 → (∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
25	24	rexbidv 3162	. . . . . 6 ⊢ (𝑣 = 𝑔 → (∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)))
26	25	abbidv 2803	. . . . 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 7527	. . . 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 1465	. . 3 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) = {𝑥 ∣ ∃𝑦 ∈ 𝑓 ∃𝑧 ∈ 𝑔 𝑥 = (𝑦𝐺𝑧)})
30	4, 9, 29	vtocl2ga 3535	. 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 3203	. . . 4 ⊢ (𝑥 = 𝑓 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑓 = (𝑦𝐺𝑧)))
33		oveq1 7375	. . . . . 6 ⊢ (𝑦 = 𝑔 → (𝑦𝐺𝑧) = (𝑔𝐺𝑧))
34	33	eqeq2d 2748	. . . . 5 ⊢ (𝑦 = 𝑔 → (𝑓 = (𝑦𝐺𝑧) ↔ 𝑓 = (𝑔𝐺𝑧)))
35		oveq2 7376	. . . . . 6 ⊢ (𝑧 = ℎ → (𝑔𝐺𝑧) = (𝑔𝐺ℎ))
36	35	eqeq2d 2748	. . . . 5 ⊢ (𝑧 = ℎ → (𝑓 = (𝑔𝐺𝑧) ↔ 𝑓 = (𝑔𝐺ℎ)))
37	34, 36	cbvrex2vw 3221	. . . 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 2807	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦𝐺𝑧)} = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}
40	30, 39	eqtrdi 2788	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  (class class class)co 7368   ∈ cmpo 7370  Qcnq 10775  Pcnp 10782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-ni 10795  df-nq 10835  df-np 10904

This theorem is referenced by:  genpelv  10923  plpv  10933  mpv  10934