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Mirrors > Home > MPE Home > Th. List > hashunlei | Structured version Visualization version GIF version |
Description: Get an upper bound on a concretely specified finite set. Induction step: union of two finite bounded sets. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
hashunlei.c | ⊢ 𝐶 = (𝐴 ∪ 𝐵) |
hashunlei.a | ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝐾) |
hashunlei.b | ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) |
hashunlei.k | ⊢ 𝐾 ∈ ℕ0 |
hashunlei.m | ⊢ 𝑀 ∈ ℕ0 |
hashunlei.n | ⊢ (𝐾 + 𝑀) = 𝑁 |
Ref | Expression |
---|---|
hashunlei | ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashunlei.c | . . 3 ⊢ 𝐶 = (𝐴 ∪ 𝐵) | |
2 | hashunlei.a | . . . . 5 ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝐾) | |
3 | 2 | simpli 485 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 485 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 9050 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 691 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2835 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6841 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
10 | hashun2 14212 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
11 | 3, 5, 10 | mp2an 691 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
12 | 9, 11 | eqbrtri 5125 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
13 | 2 | simpri 487 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 487 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
15 | hashcl 14185 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 12358 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashcl 14185 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐵) ∈ ℕ0 |
20 | 19 | nn0rei 12358 | . . . . . 6 ⊢ (♯‘𝐵) ∈ ℝ |
21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
22 | 21 | nn0rei 12358 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
24 | 23 | nn0rei 12358 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
25 | 17, 20, 22, 24 | le2addi 11652 | . . . . 5 ⊢ (((♯‘𝐴) ≤ 𝐾 ∧ (♯‘𝐵) ≤ 𝑀) → ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀)) |
26 | 13, 14, 25 | mp2an 691 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀) |
27 | hashunlei.n | . . . 4 ⊢ (𝐾 + 𝑀) = 𝑁 | |
28 | 26, 27 | breqtri 5129 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
29 | hashcl 14185 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (♯‘𝐶) ∈ ℕ0) | |
30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐶) ∈ ℕ0 |
31 | 30 | nn0rei 12358 | . . . 4 ⊢ (♯‘𝐶) ∈ ℝ |
32 | 17, 20 | readdcli 11104 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 11104 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2836 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 11218 | . . 3 ⊢ (((♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁) → (♯‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 691 | . 2 ⊢ (♯‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 472 | 1 ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cun 3907 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 Fincfn 8817 ℝcr 10984 + caddc 10988 ≤ cle 11124 ℕ0cn0 12347 ♯chash 14159 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-oadd 8384 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-dju 9771 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-n0 12348 df-xnn0 12420 df-z 12434 df-uz 12698 df-fz 13355 df-hash 14160 |
This theorem is referenced by: hashprlei 14296 hashtplei 14312 kur14lem8 33587 |
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