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Mirrors > Home > MPE Home > Th. List > psrbagaddcl | Structured version Visualization version GIF version |
Description: The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbagaddcl | ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcl 12138 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 + 𝑦) ∈ ℕ0) | |
2 | 1 | adantl 485 | . . 3 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) ∧ (𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)) → (𝑥 + 𝑦) ∈ ℕ0) |
3 | psrbag.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
4 | 3 | psrbagf 20890 | . . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0) |
6 | 3 | psrbagf 20890 | . . . 4 ⊢ (𝐺 ∈ 𝐷 → 𝐺:𝐼⟶ℕ0) |
7 | 6 | adantl 485 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺:𝐼⟶ℕ0) |
8 | simpl 486 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ 𝐷) | |
9 | 5 | ffnd 6555 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 Fn 𝐼) |
10 | 8, 9 | fndmexd 7693 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐼 ∈ V) |
11 | inidm 4142 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
12 | 2, 5, 7, 10, 10, 11 | off 7495 | . 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺):𝐼⟶ℕ0) |
13 | ovex 7255 | . . . 4 ⊢ (𝐹 ∘f + 𝐺) ∈ V | |
14 | frnnn0suppg 12161 | . . . 4 ⊢ (((𝐹 ∘f + 𝐺) ∈ V ∧ (𝐹 ∘f + 𝐺):𝐼⟶ℕ0) → ((𝐹 ∘f + 𝐺) supp 0) = (◡(𝐹 ∘f + 𝐺) “ ℕ)) | |
15 | 13, 12, 14 | sylancr 590 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) = (◡(𝐹 ∘f + 𝐺) “ ℕ)) |
16 | 3 | psrbagfsupp 20892 | . . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 finSupp 0) |
18 | 3 | psrbagfsupp 20892 | . . . . . 6 ⊢ (𝐺 ∈ 𝐷 → 𝐺 finSupp 0) |
19 | 18 | adantl 485 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 finSupp 0) |
20 | 17, 19 | fsuppunfi 9018 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin) |
21 | 0nn0 12118 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 0 ∈ ℕ0) |
23 | 00id 11020 | . . . . . 6 ⊢ (0 + 0) = 0 | |
24 | 23 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (0 + 0) = 0) |
25 | 10, 22, 5, 7, 24 | suppofssd 7954 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) |
26 | 20, 25 | ssfid 8911 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) ∈ Fin) |
27 | 15, 26 | eqeltrrd 2840 | . 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡(𝐹 ∘f + 𝐺) “ ℕ) ∈ Fin) |
28 | 3 | psrbag 20889 | . . 3 ⊢ (𝐼 ∈ V → ((𝐹 ∘f + 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘f + 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘f + 𝐺) “ ℕ) ∈ Fin))) |
29 | 10, 28 | syl 17 | . 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) ∈ 𝐷 ↔ ((𝐹 ∘f + 𝐺):𝐼⟶ℕ0 ∧ (◡(𝐹 ∘f + 𝐺) “ ℕ) ∈ Fin))) |
30 | 12, 27, 29 | mpbir2and 713 | 1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {crab 3066 Vcvv 3415 ∪ cun 3873 class class class wbr 5062 ◡ccnv 5559 “ cima 5563 ⟶wf 6385 (class class class)co 7222 ∘f cof 7476 supp csupp 7912 ↑m cmap 8517 Fincfn 8635 finSupp cfsupp 8998 0cc0 10742 + caddc 10745 ℕcn 11843 ℕ0cn0 12103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-of 7478 df-om 7654 df-1st 7770 df-2nd 7771 df-supp 7913 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-er 8400 df-map 8519 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-fsupp 8999 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-nn 11844 df-n0 12104 |
This theorem is referenced by: mplmon2mul 21040 evlslem1 21055 tdeglem3 24968 |
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