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Mirrors > Home > MPE Home > Th. List > genpelv | Structured version Visualization version GIF version |
Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 | ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) |
genp.2 | ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) |
Ref | Expression |
---|---|
genpelv | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genp.1 | . . . 4 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) | |
2 | genp.2 | . . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) | |
3 | 1, 2 | genpv 10942 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)}) |
4 | 3 | eleq2d 2824 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)})) |
5 | id 22 | . . . . . 6 ⊢ (𝐶 = (𝑔𝐺ℎ) → 𝐶 = (𝑔𝐺ℎ)) | |
6 | ovex 7395 | . . . . . 6 ⊢ (𝑔𝐺ℎ) ∈ V | |
7 | 5, 6 | eqeltrdi 2846 | . . . . 5 ⊢ (𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V) |
8 | 7 | rexlimivw 3149 | . . . 4 ⊢ (∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V) |
9 | 8 | rexlimivw 3149 | . . 3 ⊢ (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ V) |
10 | eqeq1 2741 | . . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ))) | |
11 | 10 | 2rexbidv 3214 | . . 3 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ))) |
12 | 9, 11 | elab3 3643 | . 2 ⊢ (𝐶 ∈ {𝑓 ∣ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)} ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ)) |
13 | 4, 12 | bitrdi 287 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝐶 = (𝑔𝐺ℎ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∃wrex 3074 Vcvv 3448 (class class class)co 7362 ∈ cmpo 7364 Qcnq 10795 Pcnp 10802 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-ni 10815 df-nq 10855 df-np 10924 |
This theorem is referenced by: genpprecl 10944 genpss 10947 genpnnp 10948 genpcd 10949 genpnmax 10950 genpass 10952 distrlem1pr 10968 distrlem5pr 10970 1idpr 10972 ltexprlem6 10984 reclem3pr 10992 reclem4pr 10993 |
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