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Mirrors > Home > MPE Home > Th. List > psrbagev1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of psrbagev1 20843 as of 7-Aug-2024. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
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 |
---|---|
psrbagev1OLD | ⊢ (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 19001 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
3 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
4 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
5 | 3, 4 | mulgnn0cl 18316 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
6 | 5 | 3expb 1117 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 2, 6 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | psrbagev1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
9 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
10 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
11 | 10 | psrbagfOLD 20686 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
12 | 8, 9, 11 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
13 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
14 | inidm 4125 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
15 | 7, 12, 13, 8, 8, 14 | off 7427 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺):𝐼⟶𝐶) |
16 | ovexd 7190 | . . 3 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) ∈ V) | |
17 | 12 | ffnd 6503 | . . . 4 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
18 | 13 | ffnd 6503 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | 17, 18, 8, 8 | offun 7423 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘f · 𝐺)) |
20 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
21 | 20 | fvexi 6676 | . . . 4 ⊢ 0 ∈ V |
22 | 21 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
23 | 10 | psrbagfsuppOLD 20688 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ 𝑊) → 𝐵 finSupp 0) |
24 | 9, 8, 23 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
25 | 24 | fsuppimpd 8878 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
26 | ssidd 3917 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
27 | 3, 20, 4 | mulg0 18303 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
28 | 27 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
29 | c0ex 10678 | . . . . 5 ⊢ 0 ∈ V | |
30 | 29 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
31 | 26, 28, 12, 13, 8, 30 | suppssof1 7878 | . . 3 ⊢ (𝜑 → ((𝐵 ∘f · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
32 | suppssfifsupp 8886 | . . 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 | 16, 19, 22, 25, 31, 32 | syl32anc 1375 | . 2 ⊢ (𝜑 → (𝐵 ∘f · 𝐺) finSupp 0 ) |
34 | 15, 33 | 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 3074 Vcvv 3409 ⊆ wss 3860 class class class wbr 5035 ◡ccnv 5526 “ cima 5530 Fun wfun 6333 ⟶wf 6335 ‘cfv 6339 (class class class)co 7155 ∘f cof 7408 supp csupp 7840 ↑m cmap 8421 Fincfn 8532 finSupp cfsupp 8871 0cc0 10580 ℕcn 11679 ℕ0cn0 11939 Basecbs 16546 0gc0g 16776 Mndcmnd 17982 .gcmg 18296 CMndccmn 18978 |
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 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-er 8304 df-map 8423 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-nn 11680 df-n0 11940 df-z 12026 df-uz 12288 df-fz 12945 df-seq 13424 df-0g 16778 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-mulg 18297 df-cmn 18980 |
This theorem is referenced by: psrbagev2OLD 20846 |
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