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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvvvallem | Structured version Visualization version GIF version |
Description: Lemma for evlsvvval 41624 akin to psrbagev2 21950. (Contributed by SN, 6-Mar-2025.) |
Ref | Expression |
---|---|
evlsvvvallem.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
evlsvvvallem.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsvvvallem.m | ⊢ 𝑀 = (mulGrp‘𝑆) |
evlsvvvallem.w | ⊢ ↑ = (.g‘𝑀) |
evlsvvvallem.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsvvvallem.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvvvallem.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
evlsvvvallem.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
Ref | Expression |
---|---|
evlsvvvallem | ⊢ (𝜑 → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvvvallem.m | . . 3 ⊢ 𝑀 = (mulGrp‘𝑆) | |
2 | evlsvvvallem.k | . . 3 ⊢ 𝐾 = (Base‘𝑆) | |
3 | 1, 2 | mgpbas 20035 | . 2 ⊢ 𝐾 = (Base‘𝑀) |
4 | eqid 2724 | . . 3 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
5 | 1, 4 | ringidval 20078 | . 2 ⊢ (1r‘𝑆) = (0g‘𝑀) |
6 | evlsvvvallem.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | 1 | crngmgp 20136 | . . 3 ⊢ (𝑆 ∈ CRing → 𝑀 ∈ CMnd) |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑀 ∈ CMnd) |
9 | evlsvvvallem.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
10 | evlsvvvallem.w | . . . 4 ⊢ ↑ = (.g‘𝑀) | |
11 | 6 | crngringd 20141 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring) |
12 | 1 | ringmgp 20134 | . . . . . 6 ⊢ (𝑆 ∈ Ring → 𝑀 ∈ Mnd) |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Mnd) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → 𝑀 ∈ Mnd) |
15 | evlsvvvallem.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
16 | evlsvvvallem.d | . . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
17 | 16 | psrbagf 21780 | . . . . . 6 ⊢ (𝐵 ∈ 𝐷 → 𝐵:𝐼⟶ℕ0) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
19 | 18 | ffvelcdmda 7076 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐵‘𝑣) ∈ ℕ0) |
20 | evlsvvvallem.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
21 | elmapi 8839 | . . . . . 6 ⊢ (𝐴 ∈ (𝐾 ↑m 𝐼) → 𝐴:𝐼⟶𝐾) | |
22 | 20, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴:𝐼⟶𝐾) |
23 | 22 | ffvelcdmda 7076 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐴‘𝑣) ∈ 𝐾) |
24 | 3, 10, 14, 19, 23 | mulgnn0cld 19012 | . . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣) ↑ (𝐴‘𝑣)) ∈ 𝐾) |
25 | 24 | fmpttd 7106 | . 2 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))):𝐼⟶𝐾) |
26 | 9 | mptexd 7217 | . . 3 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))) ∈ V) |
27 | fvexd 6896 | . . 3 ⊢ (𝜑 → (1r‘𝑆) ∈ V) | |
28 | 25 | ffund 6711 | . . 3 ⊢ (𝜑 → Fun (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) |
29 | 16 | psrbagfsupp 21782 | . . . 4 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0) |
30 | 15, 29 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵 finSupp 0) |
31 | ssidd 3997 | . . . . . . 7 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) | |
32 | 0zd 12567 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ) | |
33 | 18, 31, 9, 32 | suppssr 8175 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐵‘𝑣) = 0) |
34 | 33 | oveq1d 7416 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐵‘𝑣) ↑ (𝐴‘𝑣)) = (0 ↑ (𝐴‘𝑣))) |
35 | eldifi 4118 | . . . . . . 7 ⊢ (𝑣 ∈ (𝐼 ∖ (𝐵 supp 0)) → 𝑣 ∈ 𝐼) | |
36 | 35, 23 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐴‘𝑣) ∈ 𝐾) |
37 | 3, 5, 10 | mulg0 18992 | . . . . . 6 ⊢ ((𝐴‘𝑣) ∈ 𝐾 → (0 ↑ (𝐴‘𝑣)) = (1r‘𝑆)) |
38 | 36, 37 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (0 ↑ (𝐴‘𝑣)) = (1r‘𝑆)) |
39 | 34, 38 | eqtrd 2764 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐵‘𝑣) ↑ (𝐴‘𝑣)) = (1r‘𝑆)) |
40 | 39, 9 | suppss2 8180 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))) supp (1r‘𝑆)) ⊆ (𝐵 supp 0)) |
41 | 26, 27, 28, 30, 40 | fsuppsssuppgd 41557 | . 2 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))) finSupp (1r‘𝑆)) |
42 | 3, 5, 8, 9, 25, 41 | gsumcl 19825 | 1 ⊢ (𝜑 → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ∖ cdif 3937 class class class wbr 5138 ↦ cmpt 5221 ◡ccnv 5665 “ cima 5669 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 supp csupp 8140 ↑m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 0cc0 11106 ℕcn 12209 ℕ0cn0 12469 ℤcz 12555 Basecbs 17143 Σg cgsu 17385 Mndcmnd 18657 .gcmg 18985 CMndccmn 19690 mulGrpcmgp 20029 1rcur 20076 Ringcrg 20128 CRingccrg 20129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-gsum 17387 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mulg 18986 df-cntz 19223 df-cmn 19692 df-mgp 20030 df-ur 20077 df-ring 20130 df-cring 20131 |
This theorem is referenced by: evlsvvvallem2 41623 evlsvvval 41624 evlsbagval 41627 evlselv 41648 evlsmhpvvval 41656 mhphf 41658 |
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