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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.) |
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 ∈ 𝑊) |
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 18914 | . . . . 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 18236 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 6 | 3expb 1117 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | 3, 7 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
9 | psrbagev1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
10 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
11 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
12 | 11 | psrbagf 20603 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
13 | 9, 10, 12 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
14 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
15 | inidm 4145 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 8, 13, 14, 9, 9, 15 | off 7404 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
17 | ovexd 7170 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
18 | 13 | ffnd 6488 | . . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
19 | 14 | ffnd 6488 | . . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
20 | 18, 19, 9, 9, 15 | offn 7400 | . . . 4 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) Fn 𝐼) |
21 | fnfun 6423 | . . . 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 6659 | . . . 4 ⊢ 0 ∈ V |
25 | 24 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
26 | 11 | psrbagfsupp 20748 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ 𝑊) → 𝐵 finSupp 0) |
27 | 10, 9, 26 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
28 | 27 | fsuppimpd 8824 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
29 | ssidd 3938 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
30 | 4, 23, 5 | mulg0 18223 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
31 | 30 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
32 | c0ex 10624 | . . . . 5 ⊢ 0 ∈ V | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
34 | 29, 31, 13, 14, 9, 33 | suppssof1 7846 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
35 | suppssfifsupp 8832 | . . 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 1375 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
37 | 16, 36 | jca 515 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 supp csupp 7813 ↑m cmap 8389 Fincfn 8492 finSupp cfsupp 8817 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 0gc0g 16705 Mndcmnd 17903 .gcmg 18216 CMndccmn 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 df-cmn 18900 |
This theorem is referenced by: psrbagev2 20750 evlslem1 20754 |
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