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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 10351 | . . . . . . 7 ⊢ <Q ⊆ (Q × Q) | |
2 | 1 | brel 5620 | . . . . . 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 10425 | . . . . . . . 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 5073 | . . . . . . . . . . . . 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 3296 | . . . . . . 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 1536 ∈ wcel 2113 {cab 2802 ∃wrex 3142 class class class wbr 5069 (class class class)co 7159 ∈ cmpo 7161 Qcnq 10277 <Q cltq 10283 Pcnp 10284 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-ni 10297 df-nq 10337 df-ltnq 10343 df-np 10406 |
This theorem is referenced by: genpcl 10433 |
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