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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.) |
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) |
psrbagev1.i | ⊢ (𝜑 → 𝐼 ∈ V) |
Ref | Expression |
---|---|
psrbagev2 | ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘𝑓 · 𝐺)) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev1.c | . 2 ⊢ 𝐶 = (Base‘𝑇) | |
2 | psrbagev1.z | . 2 ⊢ 0 = (0g‘𝑇) | |
3 | psrbagev1.t | . 2 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
4 | psrbagev1.i | . 2 ⊢ (𝜑 → 𝐼 ∈ V) | |
5 | psrbagev1.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | psrbagev1.x | . . . 4 ⊢ · = (.g‘𝑇) | |
7 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
8 | psrbagev1.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
9 | 5, 1, 6, 2, 3, 7, 8, 4 | psrbagev1 19725 | . . 3 ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘𝑓 · 𝐺) finSupp 0 )) |
10 | 9 | simpld 476 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶) |
11 | 9 | simprd 477 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) finSupp 0 ) |
12 | 1, 2, 3, 4, 10, 11 | gsumcl 18523 | 1 ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘𝑓 · 𝐺)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 {crab 3065 Vcvv 3351 class class class wbr 4786 ◡ccnv 5248 “ cima 5252 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ∘𝑓 cof 7042 ↑𝑚 cmap 8009 Fincfn 8109 finSupp cfsupp 8431 ℕcn 11222 ℕ0cn0 11494 Basecbs 16064 0gc0g 16308 Σg cgsu 16309 .gcmg 17748 CMndccmn 18400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-fzo 12674 df-seq 13009 df-hash 13322 df-0g 16310 df-gsum 16311 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mulg 17749 df-cntz 17957 df-cmn 18402 |
This theorem is referenced by: evlslem6 19728 evlslem1 19730 |
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