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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.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-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) |
| 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 | 1 | cmnmndd 19822 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
| 3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
| 4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
| 5 | 3, 4 | mulgnn0cl 19108 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
| 6 | 5 | 3expb 1121 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
| 7 | 2, 6 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
| 8 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 9 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
| 10 | 9 | psrbagf 21938 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → 𝐵:𝐼⟶ℕ0) |
| 11 | 8, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
| 12 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
| 13 | 11 | ffnd 6737 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
| 14 | 8, 13 | fndmexd 7926 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
| 15 | inidm 4227 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
| 16 | 7, 11, 12, 14, 14, 15 | off 7715 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
| 17 | ovexd 7466 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
| 18 | 12 | ffnd 6737 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
| 19 | 13, 18, 14, 14 | offun 7711 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
| 20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
| 21 | 20 | fvexi 6920 | . . . 4 ⊢ 0 ∈ V |
| 22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
| 23 | 9 | psrbagfsupp 21939 | . . . . 5 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0) |
| 24 | 8, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
| 25 | 24 | fsuppimpd 9409 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
| 26 | ssidd 4007 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
| 27 | 3, 20, 4 | mulg0 19092 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
| 28 | 27 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
| 29 | c0ex 11255 | . . . . 5 ⊢ 0 ∈ V | |
| 30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
| 31 | 26, 28, 11, 12, 14, 30 | suppssof1 8224 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
| 32 | suppssfifsupp 9420 | . . 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 ) | |
| 33 | 17, 19, 22, 25, 31, 32 | syl32anc 1380 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
| 34 | 16, 33 | jca 511 | 1 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 0gc0g 17484 Mndcmnd 18747 .gcmg 19085 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mulg 19086 df-cmn 19800 |
| This theorem is referenced by: psrbagev2 22102 evlslem1 22106 |
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