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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.) |
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 |
---|---|
psrbagev1 | ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev1.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
2 | cmnmnd 18916 | . . . . 5 ⊢ (𝑇 ∈ CMnd → 𝑇 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
4 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
5 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
6 | 4, 5 | mulgnn0cl 18238 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 6 | 3expb 1116 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | 3, 7 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
9 | psrbagev1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) | |
10 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
11 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
12 | 11 | psrbagf 20139 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
13 | 9, 10, 12 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
14 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
15 | inidm 4194 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 8, 13, 14, 9, 9, 15 | off 7418 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
17 | ovexd 7185 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
18 | 13 | ffnd 6509 | . . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
19 | 14 | ffnd 6509 | . . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
20 | 18, 19, 9, 9, 15 | offn 7414 | . . . 4 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) Fn 𝐼) |
21 | fnfun 6447 | . . . 4 ⊢ ((𝐵 ∘f · 𝐺) Fn 𝐼 → Fun (𝐵 ∘f · 𝐺)) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
23 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
24 | 23 | fvexi 6678 | . . . 4 ⊢ 0 ∈ V |
25 | 24 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
26 | 11 | psrbagfsupp 20283 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ V) → 𝐵 finSupp 0) |
27 | 10, 9, 26 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
28 | 27 | fsuppimpd 8834 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
29 | ssidd 3989 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
30 | 4, 23, 5 | mulg0 18225 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
31 | 30 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
32 | c0ex 10629 | . . . . 5 ⊢ 0 ∈ V | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
34 | 29, 31, 13, 14, 9, 33 | suppssof1 7857 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
35 | suppssfifsupp 8842 | . . 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 ) | |
36 | 17, 22, 25, 28, 34, 35 | syl32anc 1374 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
37 | 16, 36 | jca 514 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ◡ccnv 5548 “ cima 5552 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 supp csupp 7824 ↑m cmap 8400 Fincfn 8503 finSupp cfsupp 8827 0cc0 10531 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 0gc0g 16707 Mndcmnd 17905 .gcmg 18218 CMndccmn 18900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mulg 18219 df-cmn 18902 |
This theorem is referenced by: psrbagev2 20285 evlslem1 20289 |
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