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Mirrors > Home > MPE Home > Th. List > hashsslei | Structured version Visualization version GIF version |
Description: Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
hashsslei.b | ⊢ 𝐵 ⊆ 𝐴 |
hashsslei.a | ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁) |
hashsslei.n | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
hashsslei | ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashsslei.a | . . . 4 ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁) | |
2 | 1 | simpli 485 | . . 3 ⊢ 𝐴 ∈ Fin |
3 | hashsslei.b | . . 3 ⊢ 𝐵 ⊆ 𝐴 | |
4 | ssfi 9169 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
5 | 2, 3, 4 | mp2an 691 | . 2 ⊢ 𝐵 ∈ Fin |
6 | ssdomg 8992 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
7 | 2, 3, 6 | mp2 9 | . . . 4 ⊢ 𝐵 ≼ 𝐴 |
8 | hashdom 14335 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴)) | |
9 | 5, 2, 8 | mp2an 691 | . . . 4 ⊢ ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴) |
10 | 7, 9 | mpbir 230 | . . 3 ⊢ (♯‘𝐵) ≤ (♯‘𝐴) |
11 | 1 | simpri 487 | . . 3 ⊢ (♯‘𝐴) ≤ 𝑁 |
12 | hashcl 14312 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
13 | 5, 12 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐵) ∈ ℕ0 |
14 | 13 | nn0rei 12479 | . . . 4 ⊢ (♯‘𝐵) ∈ ℝ |
15 | hashcl 14312 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 2, 15 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 12479 | . . . 4 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashsslei.n | . . . . 5 ⊢ 𝑁 ∈ ℕ0 | |
19 | 18 | nn0rei 12479 | . . . 4 ⊢ 𝑁 ∈ ℝ |
20 | 14, 17, 19 | letri 11339 | . . 3 ⊢ (((♯‘𝐵) ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≤ 𝑁) → (♯‘𝐵) ≤ 𝑁) |
21 | 10, 11, 20 | mp2an 691 | . 2 ⊢ (♯‘𝐵) ≤ 𝑁 |
22 | 5, 21 | pm3.2i 472 | 1 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3947 class class class wbr 5147 ‘cfv 6540 ≼ cdom 8933 Fincfn 8935 ≤ cle 11245 ℕ0cn0 12468 ♯chash 14286 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 |
This theorem is referenced by: kur14lem9 34143 |
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