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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 12561 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0) → (𝑥 + 𝑦) ∈ ℕ0) | |
| 2 | 1 | adantl 481 | . . 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 21938 | . . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) | 
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹:𝐼⟶ℕ0) | 
| 6 | 3 | psrbagf 21938 | . . . 4 ⊢ (𝐺 ∈ 𝐷 → 𝐺:𝐼⟶ℕ0) | 
| 7 | 6 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺:𝐼⟶ℕ0) | 
| 8 | simpl 482 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 ∈ 𝐷) | |
| 9 | 5 | ffnd 6737 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 Fn 𝐼) | 
| 10 | 8, 9 | fndmexd 7926 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐼 ∈ V) | 
| 11 | inidm 4227 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
| 12 | 2, 5, 7, 10, 10, 11 | off 7715 | . 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺):𝐼⟶ℕ0) | 
| 13 | ovex 7464 | . . . 4 ⊢ (𝐹 ∘f + 𝐺) ∈ V | |
| 14 | fcdmnn0suppg 12585 | . . . 4 ⊢ (((𝐹 ∘f + 𝐺) ∈ V ∧ (𝐹 ∘f + 𝐺):𝐼⟶ℕ0) → ((𝐹 ∘f + 𝐺) supp 0) = (◡(𝐹 ∘f + 𝐺) “ ℕ)) | |
| 15 | 13, 12, 14 | sylancr 587 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) = (◡(𝐹 ∘f + 𝐺) “ ℕ)) | 
| 16 | 3 | psrbagfsupp 21939 | . . . . . 6 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) | 
| 17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐹 finSupp 0) | 
| 18 | 3 | psrbagfsupp 21939 | . . . . . 6 ⊢ (𝐺 ∈ 𝐷 → 𝐺 finSupp 0) | 
| 19 | 18 | adantl 481 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 𝐺 finSupp 0) | 
| 20 | 17, 19 | fsuppunfi 9428 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin) | 
| 21 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → 0 ∈ ℕ0) | 
| 23 | 00id 11436 | . . . . . 6 ⊢ (0 + 0) = 0 | |
| 24 | 23 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (0 + 0) = 0) | 
| 25 | 10, 22, 5, 7, 24 | suppofssd 8228 | . . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))) | 
| 26 | 20, 25 | ssfid 9301 | . . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → ((𝐹 ∘f + 𝐺) supp 0) ∈ Fin) | 
| 27 | 15, 26 | eqeltrrd 2842 | . 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (◡(𝐹 ∘f + 𝐺) “ ℕ) ∈ Fin) | 
| 28 | 3 | psrbag 21937 | . . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∪ cun 3949 class class class wbr 5143 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 (class class class)co 7431 ∘f cof 7695 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 0cc0 11155 + caddc 11158 ℕcn 12266 ℕ0cn0 12526 | 
| 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-rep 5279 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 | 
| 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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-nn 12267 df-n0 12527 | 
| This theorem is referenced by: psrbagleadd1 21948 mplmon2mul 22093 evlslem1 22106 psdcl 22165 psdmplcl 22166 psdadd 22167 psdvsca 22168 psdmul 22170 psdmvr 22173 tdeglem3 26098 | 
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