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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsummptres2 | Structured version Visualization version GIF version |
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 22-Jun-2024.) |
Ref | Expression |
---|---|
gsummptres2.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptres2.z | ⊢ 0 = (0g‘𝐺) |
gsummptres2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptres2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptres2.0 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) |
gsummptres2.1 | ⊢ (𝜑 → 𝑆 ∈ Fin) |
gsummptres2.y | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) |
gsummptres2.2 | ⊢ (𝜑 → 𝑆 ⊆ 𝐴) |
Ref | Expression |
---|---|
gsummptres2 | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptres2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptres2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2735 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | gsummptres2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
5 | gsummptres2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | gsummptres2.y | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
7 | 5 | mptexd 7244 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V) |
8 | funmpt 6606 | . . . . 5 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → Fun (𝑥 ∈ 𝐴 ↦ 𝑌)) |
10 | 2 | fvexi 6921 | . . . . 5 ⊢ 0 ∈ V |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
12 | gsummptres2.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin) | |
13 | gsummptres2.0 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) | |
14 | 13, 5 | suppss2 8224 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆) |
15 | suppssfifsupp 9418 | . . . 4 ⊢ ((((𝑥 ∈ 𝐴 ↦ 𝑌) ∈ V ∧ Fun (𝑥 ∈ 𝐴 ↦ 𝑌) ∧ 0 ∈ V) ∧ (𝑆 ∈ Fin ∧ ((𝑥 ∈ 𝐴 ↦ 𝑌) supp 0 ) ⊆ 𝑆)) → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) | |
16 | 7, 9, 11, 12, 14, 15 | syl32anc 1377 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) |
17 | disjdif 4478 | . . . 4 ⊢ (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅ | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑆 ∩ (𝐴 ∖ 𝑆)) = ∅) |
19 | gsummptres2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐴) | |
20 | undif 4488 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 ↔ (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) | |
21 | 19, 20 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ (𝐴 ∖ 𝑆)) = 𝐴) |
22 | 21 | eqcomd 2741 | . . 3 ⊢ (𝜑 → 𝐴 = (𝑆 ∪ (𝐴 ∖ 𝑆))) |
23 | 1, 2, 3, 4, 5, 6, 16, 18, 22 | gsumsplit2 19962 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)))) |
24 | 13 | mpteq2dva 5248 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌) = (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) |
25 | 24 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 ))) |
26 | 4 | cmnmndd 19837 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
27 | 5 | difexd 5337 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝑆) ∈ V) |
28 | 2 | gsumz 18862 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∖ 𝑆) ∈ V) → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
29 | 26, 27, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 0 )) = 0 ) |
30 | 25, 29 | eqtrd 2775 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌)) = 0 ) |
31 | 30 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺)(𝐺 Σg (𝑥 ∈ (𝐴 ∖ 𝑆) ↦ 𝑌))) = ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 )) |
32 | 6 | ralrimiva 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵) |
33 | ssralv 4064 | . . . . 5 ⊢ (𝑆 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 𝑌 ∈ 𝐵 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵)) | |
34 | 19, 32, 33 | sylc 65 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑌 ∈ 𝐵) |
35 | 1, 4, 12, 34 | gsummptcl 20000 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) |
36 | 1, 3, 2 | mndrid 18781 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌)) ∈ 𝐵) → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
37 | 26, 35, 36 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))(+g‘𝐺) 0 ) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
38 | 23, 31, 37 | 3eqtrd 2779 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σg (𝑥 ∈ 𝑆 ↦ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ↦ cmpt 5231 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 supp csupp 8184 Fincfn 8984 finSupp cfsupp 9399 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Σg cgsu 17487 Mndcmnd 18760 CMndccmn 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-cntz 19348 df-cmn 19815 |
This theorem is referenced by: gsumfs2d 33041 elrgspnlem4 33235 elrspunidl 33436 gsummoncoe1fzo 33598 |
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