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Mirrors > Home > MPE Home > Th. List > genpcd | Structured version Visualization version GIF version |
Description: Downward closure of an operation 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) |
genpcd.2 | ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵))) |
Ref | Expression |
---|---|
genpcd | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 10342 | . . . . . . 7 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 5611 | . . . . . 6 ⊢ (𝑥 <Q 𝑓 → (𝑥 ∈ Q ∧ 𝑓 ∈ Q)) |
3 | 2 | simpld 497 | . . . . 5 ⊢ (𝑥 <Q 𝑓 → 𝑥 ∈ Q) |
4 | genp.1 | . . . . . . . . 9 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) | |
5 | genp.2 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) | |
6 | 4, 5 | genpelv 10416 | . . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ))) |
7 | 6 | adantr 483 | . . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ))) |
8 | breq2 5062 | . . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 ↔ 𝑥 <Q (𝑔𝐺ℎ))) | |
9 | 8 | biimpd 231 | . . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 → 𝑥 <Q (𝑔𝐺ℎ))) |
10 | genpcd.2 | . . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵))) | |
11 | 9, 10 | sylan9r 511 | . . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) ∧ 𝑓 = (𝑔𝐺ℎ)) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))) |
12 | 11 | exp31 422 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))) |
13 | 12 | an4s 658 | . . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))) |
14 | 13 | impancom 454 | . . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))) |
15 | 14 | rexlimdvv 3293 | . . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
16 | 7, 15 | sylbid 242 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
17 | 16 | ex 415 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ Q → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))) |
18 | 3, 17 | syl5 34 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵))))) |
19 | 18 | com34 91 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <Q 𝑓 → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑥 ∈ (𝐴𝐹𝐵))))) |
20 | 19 | pm2.43d 53 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 <Q 𝑓 → (𝑓 ∈ (𝐴𝐹𝐵) → 𝑥 ∈ (𝐴𝐹𝐵)))) |
21 | 20 | com23 86 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∃wrex 3139 class class class wbr 5058 (class class class)co 7150 ∈ cmpo 7152 Qcnq 10268 <Q cltq 10274 Pcnp 10275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-ni 10288 df-nq 10328 df-ltnq 10334 df-np 10397 |
This theorem is referenced by: genpcl 10424 |
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