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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 487 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 487 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 8769 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 691 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2886 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6648 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
10 | hashun2 13740 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
11 | 3, 5, 10 | mp2an 691 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
12 | 9, 11 | eqbrtri 5051 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
13 | 2 | simpri 489 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 489 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
15 | hashcl 13713 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 11896 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashcl 13713 | . . . . . . . 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 11192 | . . . . 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 5055 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
29 | hashcl 13713 | . . . . . 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 10645 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 10645 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2887 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 10758 | . . 3 ⊢ (((♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁) → (♯‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 691 | . 2 ⊢ (♯‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 474 | 1 ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℝcr 10525 + caddc 10529 ≤ cle 10665 ℕ0cn0 11885 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: hashprlei 13822 hashtplei 13838 kur14lem8 32573 |
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