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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 482 | . . . 4 ⊢ 𝐴 ∈ Fin |
4 | hashunlei.b | . . . . 5 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑀) | |
5 | 4 | simpli 482 | . . . 4 ⊢ 𝐵 ∈ Fin |
6 | unfi 9195 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
7 | 3, 5, 6 | mp2an 690 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
8 | 1, 7 | eqeltri 2821 | . 2 ⊢ 𝐶 ∈ Fin |
9 | 1 | fveq2i 6895 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
10 | hashun2 14374 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
11 | 3, 5, 10 | mp2an 690 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
12 | 9, 11 | eqbrtri 5164 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
13 | 2 | simpri 484 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
14 | 4 | simpri 484 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
15 | hashcl 14347 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 12513 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashcl 14347 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐵) ∈ ℕ0 |
20 | 19 | nn0rei 12513 | . . . . . 6 ⊢ (♯‘𝐵) ∈ ℝ |
21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
22 | 21 | nn0rei 12513 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
24 | 23 | nn0rei 12513 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
25 | 17, 20, 22, 24 | le2addi 11807 | . . . . 5 ⊢ (((♯‘𝐴) ≤ 𝐾 ∧ (♯‘𝐵) ≤ 𝑀) → ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀)) |
26 | 13, 14, 25 | mp2an 690 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ (𝐾 + 𝑀) |
27 | hashunlei.n | . . . 4 ⊢ (𝐾 + 𝑀) = 𝑁 | |
28 | 26, 27 | breqtri 5168 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
29 | hashcl 14347 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (♯‘𝐶) ∈ ℕ0) | |
30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐶) ∈ ℕ0 |
31 | 30 | nn0rei 12513 | . . . 4 ⊢ (♯‘𝐶) ∈ ℝ |
32 | 17, 20 | readdcli 11259 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
33 | 22, 24 | readdcli 11259 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
34 | 27, 33 | eqeltrri 2822 | . . . 4 ⊢ 𝑁 ∈ ℝ |
35 | 31, 32, 34 | letri 11373 | . . 3 ⊢ (((♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁) → (♯‘𝐶) ≤ 𝑁) |
36 | 12, 28, 35 | mp2an 690 | . 2 ⊢ (♯‘𝐶) ≤ 𝑁 |
37 | 8, 36 | pm3.2i 469 | 1 ⊢ (𝐶 ∈ Fin ∧ (♯‘𝐶) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3937 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 ℝcr 11137 + caddc 11141 ≤ cle 11279 ℕ0cn0 12502 ♯chash 14321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322 |
This theorem is referenced by: hashprlei 14461 hashtplei 14477 kur14lem8 34880 |
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