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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 484 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 484 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 8773 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 688 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2906 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6666 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
10 | hashun2 13732 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
11 | 3, 5, 10 | mp2an 688 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
12 | 9, 11 | eqbrtri 5078 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
13 | 2 | simpri 486 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 486 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
15 | hashcl 13705 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 11896 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashcl 13705 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐵) ∈ ℕ0 |
20 | 19 | nn0rei 11896 | . . . . . 6 ⊢ (♯‘𝐵) ∈ ℝ |
21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
22 | 21 | nn0rei 11896 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
24 | 23 | nn0rei 11896 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
25 | 17, 20, 22, 24 | le2addi 11191 | . . . . 5 ⊢ (((♯‘𝐴) ≤ 𝐾 ∧ (♯‘𝐵) ≤ 𝑀) → ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀)) |
26 | 13, 14, 25 | mp2an 688 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀) |
27 | hashunlei.n | . . . 4 ⊢ (𝐾 + 𝑀) = 𝑁 | |
28 | 26, 27 | breqtri 5082 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
29 | hashcl 13705 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (♯‘𝐶) ∈ ℕ0) | |
30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐶) ∈ ℕ0 |
31 | 30 | nn0rei 11896 | . . . 4 ⊢ (♯‘𝐶) ∈ ℝ |
32 | 17, 20 | readdcli 10644 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 10644 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2907 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 10757 | . . 3 ⊢ (((♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁) → (♯‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 688 | . 2 ⊢ (♯‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 471 | 1 ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℝcr 10524 + caddc 10528 ≤ cle 10664 ℕ0cn0 11885 ♯chash 13678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 |
This theorem is referenced by: hashprlei 13814 hashtplei 13830 kur14lem8 32357 |
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