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Mirrors > Home > MPE Home > Th. List > psrbagev2 | Structured version Visualization version GIF version |
Description: Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbagev2.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrbagev2.c | ⊢ 𝐶 = (Base‘𝑇) |
psrbagev2.x | ⊢ · = (.g‘𝑇) |
psrbagev2.t | ⊢ (𝜑 → 𝑇 ∈ CMnd) |
psrbagev2.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
psrbagev2.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
Ref | Expression |
---|---|
psrbagev2 | ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev2.c | . 2 ⊢ 𝐶 = (Base‘𝑇) | |
2 | eqid 2738 | . 2 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
3 | psrbagev2.t | . 2 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
4 | ovexd 7205 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
5 | psrbagev2.d | . . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | psrbagev2.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
7 | psrbagev2.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
8 | psrbagev2.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
9 | 5, 1, 6, 2, 3, 7, 8 | psrbagev1 20889 | . . . . 5 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp (0g‘𝑇))) |
10 | 9 | simpld 498 | . . . 4 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
11 | 10 | ffnd 6505 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) Fn 𝐼) |
12 | 4, 11 | fndmexd 7637 | . 2 ⊢ (𝜑 → 𝐼 ∈ V) |
13 | 9 | simprd 499 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp (0g‘𝑇)) |
14 | 1, 2, 3, 12, 10, 13 | gsumcl 19154 | 1 ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3057 Vcvv 3398 class class class wbr 5030 ◡ccnv 5524 “ cima 5528 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∘f cof 7423 ↑m cmap 8437 Fincfn 8555 finSupp cfsupp 8906 ℕcn 11716 ℕ0cn0 11976 Basecbs 16586 0gc0g 16816 Σg cgsu 16817 .gcmg 18342 CMndccmn 19024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-0g 16818 df-gsum 16819 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mulg 18343 df-cntz 18565 df-cmn 19026 |
This theorem is referenced by: evlslem6 20895 evlslem1 20896 evlsbagval 39854 |
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