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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 18297 | . . . . 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 17648 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 6 | 3expb 1113 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | 3, 7 | sylan 489 | . . 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 19456 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
13 | 9, 10, 12 | syl2anc 696 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
14 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
15 | inidm 3898 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 8, 13, 14, 9, 9, 15 | off 6997 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶) |
17 | ovexd 6763 | . . 3 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) ∈ V) | |
18 | ffn 6126 | . . . . . 6 ⊢ (𝐵:𝐼⟶ℕ0 → 𝐵 Fn 𝐼) | |
19 | 13, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
20 | ffn 6126 | . . . . . 6 ⊢ (𝐺:𝐼⟶𝐶 → 𝐺 Fn 𝐼) | |
21 | 14, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
22 | 19, 21, 9, 9, 15 | offn 6993 | . . . 4 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) Fn 𝐼) |
23 | fnfun 6069 | . . . 4 ⊢ ((𝐵 ∘𝑓 · 𝐺) Fn 𝐼 → Fun (𝐵 ∘𝑓 · 𝐺)) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘𝑓 · 𝐺)) |
25 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
26 | fvex 6282 | . . . . 5 ⊢ (0g‘𝑇) ∈ V | |
27 | 25, 26 | eqeltri 2767 | . . . 4 ⊢ 0 ∈ V |
28 | 27 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
29 | 11 | psrbagfsupp 19600 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ V) → 𝐵 finSupp 0) |
30 | 10, 9, 29 | syl2anc 696 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
31 | 30 | fsuppimpd 8366 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
32 | ssid 3698 | . . . . 5 ⊢ (𝐵 supp 0) ⊆ (𝐵 supp 0) | |
33 | 32 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) |
34 | 4, 25, 5 | mulg0 17636 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
35 | 34 | adantl 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
36 | c0ex 10115 | . . . . 5 ⊢ 0 ∈ V | |
37 | 36 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
38 | 33, 35, 13, 14, 9, 37 | suppssof1 7416 | . . 3 ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
39 | suppssfifsupp 8374 | . . 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 ) | |
40 | 17, 24, 28, 31, 38, 39 | syl32anc 1415 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) finSupp 0 ) |
41 | 16, 40 | jca 555 | 1 ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘𝑓 · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1564 ∈ wcel 2071 {crab 2986 Vcvv 3272 ⊆ wss 3648 class class class wbr 4728 ◡ccnv 5185 “ cima 5189 Fun wfun 5963 Fn wfn 5964 ⟶wf 5965 ‘cfv 5969 (class class class)co 6733 ∘𝑓 cof 6980 supp csupp 7383 ↑𝑚 cmap 7942 Fincfn 8040 finSupp cfsupp 8359 0cc0 10017 ℕcn 11101 ℕ0cn0 11373 Basecbs 15948 0gc0g 16191 Mndcmnd 17384 .gcmg 17630 CMndccmn 18282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-inf2 8619 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-of 6982 df-om 7151 df-1st 7253 df-2nd 7254 df-supp 7384 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-er 7830 df-map 7944 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-fsupp 8360 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-n0 11374 df-z 11459 df-uz 11769 df-fz 12409 df-seq 12885 df-0g 16193 df-mgm 17332 df-sgrp 17374 df-mnd 17385 df-mulg 17631 df-cmn 18284 |
This theorem is referenced by: psrbagev2 19602 evlslem1 19606 |
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