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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 9211 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin) | |
| 7 | 3, 5, 6 | mp2an 692 | . . 3 ⊢ (𝐴 ∪ 𝐵) ∈ Fin |
| 8 | 1, 7 | eqeltri 2837 | . 2 ⊢ 𝐶 ∈ Fin |
| 9 | 1 | fveq2i 6909 | . . . 4 ⊢ (♯‘𝐶) = (♯‘(𝐴 ∪ 𝐵)) |
| 10 | hashun2 14422 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵))) | |
| 11 | 3, 5, 10 | mp2an 692 | . . . 4 ⊢ (♯‘(𝐴 ∪ 𝐵)) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
| 12 | 9, 11 | eqbrtri 5164 | . . 3 ⊢ (♯‘𝐶) ≤ ((♯‘𝐴) + (♯‘𝐵)) |
| 13 | 2 | simpri 485 | . . . . 5 ⊢ (♯‘𝐴) ≤ 𝐾 |
| 14 | 4 | simpri 485 | . . . . 5 ⊢ (♯‘𝐵) ≤ 𝑀 |
| 15 | hashcl 14395 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
| 16 | 3, 15 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐴) ∈ ℕ0 |
| 17 | 16 | nn0rei 12537 | . . . . . 6 ⊢ (♯‘𝐴) ∈ ℝ |
| 18 | hashcl 14395 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
| 19 | 5, 18 | ax-mp 5 | . . . . . . 7 ⊢ (♯‘𝐵) ∈ ℕ0 |
| 20 | 19 | nn0rei 12537 | . . . . . 6 ⊢ (♯‘𝐵) ∈ ℝ |
| 21 | hashunlei.k | . . . . . . 7 ⊢ 𝐾 ∈ ℕ0 | |
| 22 | 21 | nn0rei 12537 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
| 23 | hashunlei.m | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
| 24 | 23 | nn0rei 12537 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
| 25 | 17, 20, 22, 24 | le2addi 11826 | . . . . 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 5168 | . . 3 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝑁 |
| 29 | hashcl 14395 | . . . . . 6 ⊢ (𝐶 ∈ Fin → (♯‘𝐶) ∈ ℕ0) | |
| 30 | 8, 29 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐶) ∈ ℕ0 |
| 31 | 30 | nn0rei 12537 | . . . 4 ⊢ (♯‘𝐶) ∈ ℝ |
| 32 | 17, 20 | readdcli 11276 | . . . 4 ⊢ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ |
| 33 | 22, 24 | readdcli 11276 | . . . . 5 ⊢ (𝐾 + 𝑀) ∈ ℝ |
| 34 | 27, 33 | eqeltrri 2838 | . . . 4 ⊢ 𝑁 ∈ ℝ |
| 35 | 31, 32, 34 | letri 11390 | . . 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 1540 ∈ wcel 2108 ∪ cun 3949 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℝcr 11154 + caddc 11158 ≤ cle 11296 ℕ0cn0 12526 ♯chash 14369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 |
| This theorem is referenced by: hashprlei 14507 hashtplei 14523 kur14lem8 35218 |
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