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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 |
---|---|
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) |
psrbagev2.i | ⊢ (𝜑 → 𝐼 ∈ V) |
Ref | Expression |
---|---|
psrbagev2 | ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev2.c | . 2 ⊢ 𝐶 = (Base‘𝑇) | |
2 | eqid 2823 | . 2 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
3 | psrbagev2.t | . 2 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
4 | psrbagev2.i | . 2 ⊢ (𝜑 → 𝐼 ∈ V) | |
5 | psrbagev2.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | psrbagev2.x | . . . 4 ⊢ · = (.g‘𝑇) | |
7 | psrbagev2.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
8 | psrbagev2.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
9 | 5, 1, 6, 2, 3, 7, 8, 4 | psrbagev1 20292 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp (0g‘𝑇))) |
10 | 9 | simpld 497 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
11 | 9 | simprd 498 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp (0g‘𝑇)) |
12 | 1, 2, 3, 4, 10, 11 | gsumcl 19037 | 1 ⊢ (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 class class class wbr 5068 ◡ccnv 5556 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 ↑m cmap 8408 Fincfn 8511 finSupp cfsupp 8835 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 0gc0g 16715 Σg cgsu 16716 .gcmg 18226 CMndccmn 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mulg 18227 df-cntz 18449 df-cmn 18910 |
This theorem is referenced by: evlslem6 20296 evlslem1 20297 |
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