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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 19836 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
5 | 3, 4 | mulgnn0cl 19120 | . . . . 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 21955 | . . . 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 6737 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
14 | 8, 13 | fndmexd 7926 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
15 | inidm 4234 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 7, 11, 12, 14, 14, 15 | off 7714 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
17 | ovexd 7465 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
18 | 12 | ffnd 6737 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | 13, 18, 14, 14 | offun 7710 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
21 | 20 | fvexi 6920 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
23 | 9 | psrbagfsupp 21956 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0) |
24 | 8, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
25 | 24 | fsuppimpd 9406 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
26 | ssidd 4018 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
27 | 3, 20, 4 | mulg0 19104 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
29 | c0ex 11252 | . . . . 5 ⊢ 0 ∈ V | |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
31 | 26, 28, 11, 12, 14, 30 | suppssof1 8222 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
32 | suppssfifsupp 9417 | . . 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 511 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ⊆ wss 3962 class class class wbr 5147 ◡ccnv 5687 “ cima 5691 Fun wfun 6556 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 supp csupp 8183 ↑m cmap 8864 Fincfn 8983 finSupp cfsupp 9398 0cc0 11152 ℕcn 12263 ℕ0cn0 12523 Basecbs 17244 0gc0g 17485 Mndcmnd 18759 .gcmg 19097 CMndccmn 19812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-seq 14039 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mulg 19098 df-cmn 19814 |
This theorem is referenced by: psrbagev2 22119 evlslem1 22123 |
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