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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumfsupp | Structured version Visualization version GIF version |
Description: A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.) |
Ref | Expression |
---|---|
gsumfsupp.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumfsupp.z | ⊢ 0 = (0g‘𝐺) |
gsumfsupp.s | ⊢ 𝐼 = (𝐹 supp 0 ) |
gsumfsupp.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumfsupp.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumfsupp.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumfsupp.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumfsupp | ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumfsupp.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumfsupp.z | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | gsumfsupp.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumfsupp.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | gsumfsupp.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
6 | gsumfsupp.s | . . . 4 ⊢ 𝐼 = (𝐹 supp 0 ) | |
7 | 6 | eqimss2i 4043 | . . 3 ⊢ (𝐹 supp 0 ) ⊆ 𝐼 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐼) |
9 | gsumfsupp.w | . 2 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
10 | 1, 2, 3, 4, 5, 8, 9 | gsumres 19882 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 class class class wbr 5152 ↾ cres 5684 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 supp csupp 8173 finSupp cfsupp 9395 Basecbs 17189 0gc0g 17430 Σg cgsu 17431 CMndccmn 19749 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-0g 17432 df-gsum 17433 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-cntz 19282 df-cmn 19751 |
This theorem is referenced by: (None) |
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