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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 483 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 483 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 9209 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 692 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2834 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6909 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
10 | hashun2 14418 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
11 | 3, 5, 10 | mp2an 692 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
12 | 9, 11 | eqbrtri 5168 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
13 | 2 | simpri 485 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 485 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
15 | hashcl 14391 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 12534 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashcl 14391 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐵) ∈ ℕ0 |
20 | 19 | nn0rei 12534 | . . . . . 6 ⊢ (♯‘𝐵) ∈ ℝ |
21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
22 | 21 | nn0rei 12534 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
24 | 23 | nn0rei 12534 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
25 | 17, 20, 22, 24 | le2addi 11823 | . . . . 5 ⊢ (((♯‘𝐴) ≤ 𝐾 ∧ (♯‘𝐵) ≤ 𝑀) → ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀)) |
26 | 13, 14, 25 | mp2an 692 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀) |
27 | hashunlei.n | . . . 4 ⊢ (𝐾 + 𝑀) = 𝑁 | |
28 | 26, 27 | breqtri 5172 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
29 | hashcl 14391 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (♯‘𝐶) ∈ ℕ0) | |
30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐶) ∈ ℕ0 |
31 | 30 | nn0rei 12534 | . . . 4 ⊢ (♯‘𝐶) ∈ ℝ |
32 | 17, 20 | readdcli 11273 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 11273 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2835 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 11387 | . . 3 ⊢ (((♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁) → (♯‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 692 | . 2 ⊢ (♯‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 470 | 1 ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∪ cun 3960 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ℝcr 11151 + caddc 11155 ≤ cle 11293 ℕ0cn0 12523 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 |
This theorem is referenced by: hashprlei 14503 hashtplei 14519 kur14lem8 35197 |
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