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Mirrors > Home > MPE Home > Th. List > psrbagev1 | Structured version Visualization version GIF version |
Description: A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbagev1.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrbagev1.c | ⊢ 𝐶 = (Base‘𝑇) |
psrbagev1.x | ⊢ · = (.g‘𝑇) |
psrbagev1.z | ⊢ 0 = (0g‘𝑇) |
psrbagev1.t | ⊢ (𝜑 → 𝑇 ∈ CMnd) |
psrbagev1.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
psrbagev1.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
Ref | Expression |
---|---|
psrbagev1 | ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev1.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
2 | 1 | cmnmndd 19397 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
5 | 3, 4 | mulgnn0cl 18708 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
6 | 5 | 3expb 1119 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 2, 6 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
9 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
10 | 9 | psrbagf 21109 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → 𝐵:𝐼⟶ℕ0) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
12 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
13 | 11 | ffnd 6594 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
14 | 8, 13 | fndmexd 7744 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
15 | inidm 4153 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 7, 11, 12, 14, 14, 15 | off 7542 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
17 | ovexd 7303 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
18 | 12 | ffnd 6594 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | 13, 18, 14, 14 | offun 7538 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
21 | 20 | fvexi 6781 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
23 | 9 | psrbagfsupp 21111 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0) |
24 | 8, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
25 | 24 | fsuppimpd 9123 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
26 | ssidd 3944 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
27 | 3, 20, 4 | mulg0 18695 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
28 | 27 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
29 | c0ex 10957 | . . . . 5 ⊢ 0 ∈ V | |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
31 | 26, 28, 11, 12, 14, 30 | suppssof1 8003 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
32 | suppssfifsupp 9131 | . . 3 ⊢ ((((𝐵 ∘f · 𝐺) ∈ V ∧ Fun (𝐵 ∘f · 𝐺) ∧ 0 ∈ V) ∧ ((𝐵 supp 0) ∈ Fin ∧ ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0))) → (𝐵 ∘f · 𝐺) finSupp 0 ) | |
33 | 17, 19, 22, 25, 31, 32 | syl32anc 1377 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
34 | 16, 33 | jca 512 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3430 ⊆ wss 3887 class class class wbr 5074 ◡ccnv 5584 “ cima 5588 Fun wfun 6421 ⟶wf 6423 ‘cfv 6427 (class class class)co 7268 ∘f cof 7522 supp csupp 7965 ↑m cmap 8603 Fincfn 8721 finSupp cfsupp 9116 0cc0 10859 ℕcn 11961 ℕ0cn0 12221 Basecbs 16900 0gc0g 17138 Mndcmnd 18373 .gcmg 18688 CMndccmn 19374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-seq 13710 df-0g 17140 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mulg 18689 df-cmn 19376 |
This theorem is referenced by: psrbagev2 21275 evlslem1 21280 |
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