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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 19759 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
5 | 3, 4 | mulgnn0cl 19045 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
6 | 5 | 3expb 1118 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 2, 6 | sylan 579 | . . 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 21851 | . . . 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 6723 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
14 | 8, 13 | fndmexd 7912 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
15 | inidm 4219 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 7, 11, 12, 14, 14, 15 | off 7703 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
17 | ovexd 7455 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
18 | 12 | ffnd 6723 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | 13, 18, 14, 14 | offun 7699 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
21 | 20 | fvexi 6911 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
23 | 9 | psrbagfsupp 21853 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0) |
24 | 8, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
25 | 24 | fsuppimpd 9394 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
26 | ssidd 4003 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
27 | 3, 20, 4 | mulg0 19030 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
28 | 27 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
29 | c0ex 11239 | . . . . 5 ⊢ 0 ∈ V | |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
31 | 26, 28, 11, 12, 14, 30 | suppssof1 8205 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
32 | suppssfifsupp 9404 | . . 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 1376 | . 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 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 ⊆ wss 3947 class class class wbr 5148 ◡ccnv 5677 “ cima 5681 Fun wfun 6542 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∘f cof 7683 supp csupp 8165 ↑m cmap 8845 Fincfn 8964 finSupp cfsupp 9386 0cc0 11139 ℕcn 12243 ℕ0cn0 12503 Basecbs 17180 0gc0g 17421 Mndcmnd 18694 .gcmg 19023 CMndccmn 19735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-seq 14000 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mulg 19024 df-cmn 19737 |
This theorem is referenced by: psrbagev2 22023 evlslem1 22028 |
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