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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 473 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (#‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 473 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 8112 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 704 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2684 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6106 | . . . 4 ⊢ (#‘𝐶) = (#‘(𝐴 ∪ 𝐵)) |
10 | hashun2 13033 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) ≤ ((#‘𝐴) + (#‘𝐵))) | |
11 | 3, 5, 10 | mp2an 704 | . . . 4 ⊢ (#‘(𝐴 ∪ 𝐵)) ≤ ((#‘𝐴) + (#‘𝐵)) |
12 | 9, 11 | eqbrtri 4604 | . . 3 ⊢ (#‘𝐶) ≤ ((#‘𝐴) + (#‘𝐵)) |
13 | 2 | simpri 477 | . . . . 5 ⊢ (#‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 477 | . . . . 5 ⊢ (#‘𝐵) ≤ 𝑀 |
15 | hashcl 13009 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (#‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 11180 | . . . . . 6 ⊢ (#‘𝐴) ∈ ℝ |
18 | hashcl 13009 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0) | |
19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (#‘𝐵) ∈ ℕ0 |
20 | 19 | nn0rei 11180 | . . . . . 6 ⊢ (#‘𝐵) ∈ ℝ |
21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
22 | 21 | nn0rei 11180 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
24 | 23 | nn0rei 11180 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
25 | 17, 20, 22, 24 | le2addi 10470 | . . . . 5 ⊢ (((#‘𝐴) ≤ 𝐾 ∧ (#‘𝐵) ≤ 𝑀) → ((#‘𝐴) + (#‘𝐵)) ≤ (𝐾 + 𝑀)) |
26 | 13, 14, 25 | mp2an 704 | . . . 4 ⊢ ((#‘𝐴) + (#‘𝐵)) ≤ (𝐾 + 𝑀) |
27 | hashunlei.n | . . . 4 ⊢ (𝐾 + 𝑀) = 𝑁 | |
28 | 26, 27 | breqtri 4608 | . . 3 ⊢ ((#‘𝐴) + (#‘𝐵)) ≤ 𝑁 |
29 | hashcl 13009 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (#‘𝐶) ∈ ℕ0) | |
30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (#‘𝐶) ∈ ℕ0 |
31 | 30 | nn0rei 11180 | . . . 4 ⊢ (#‘𝐶) ∈ ℝ |
32 | 17, 20 | readdcli 9932 | . . . 4 ⊢ ((#‘𝐴) + (#‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 9932 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2685 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 10045 | . . 3 ⊢ (((#‘𝐶) ≤ ((#‘𝐴) + (#‘𝐵)) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝑁) → (#‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 704 | . 2 ⊢ (#‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 470 | 1 ⊢ (𝐶 ∈ Fin ∧ (#‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 + caddc 9818 ≤ cle 9954 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: hashprlei 13107 hashtplei 13120 kur14lem8 30449 |
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