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Mirrors > Home > MPE Home > Th. List > gsumxp2 | Structured version Visualization version GIF version |
Description: Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.) |
Ref | Expression |
---|---|
gsumxp2.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumxp2.z | ⊢ 0 = (0g‘𝐺) |
gsumxp2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumxp2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumxp2.r | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
gsumxp2.f | ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) |
gsumxp2.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumxp2 | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumxp2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumxp2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumxp2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumxp2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | gsumxp2.r | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
6 | gsumxp2.f | . . . 4 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) | |
7 | 6 | fovcdmda 7525 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (𝑗𝐹𝑘) ∈ 𝐵) |
8 | gsumxp2.w | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
9 | 8 | fsuppimpd 9312 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
10 | simpl 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝜑) | |
11 | opelxpi 5670 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → 〈𝑗, 𝑘〉 ∈ (𝐴 × 𝐶)) | |
12 | 11 | ad2antlr 725 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 × 𝐶)) |
13 | simpr 485 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
14 | 12, 13 | eldifd 3921 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ ((𝐴 × 𝐶) ∖ (𝐹 supp 0 ))) |
15 | ssidd 3967 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
16 | 4, 5 | xpexd 7685 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
17 | 2 | fvexi 6856 | . . . . . . . . 9 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ V) |
19 | 6, 15, 16, 18 | suppssr 8127 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ ((𝐴 × 𝐶) ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
20 | 10, 14, 19 | syl2an2r 683 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
21 | 20 | ex 413 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ) → (𝐹‘〈𝑗, 𝑘〉) = 0 )) |
22 | df-br 5106 | . . . . . 6 ⊢ (𝑗(𝐹 supp 0 )𝑘 ↔ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
23 | 22 | notbii 319 | . . . . 5 ⊢ (¬ 𝑗(𝐹 supp 0 )𝑘 ↔ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) |
24 | df-ov 7360 | . . . . . 6 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
25 | 24 | eqeq1i 2741 | . . . . 5 ⊢ ((𝑗𝐹𝑘) = 0 ↔ (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
26 | 21, 23, 25 | 3imtr4g 295 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 𝑗(𝐹 supp 0 )𝑘 → (𝑗𝐹𝑘) = 0 )) |
27 | 26 | impr 455 | . . 3 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗(𝐹 supp 0 )𝑘)) → (𝑗𝐹𝑘) = 0 ) |
28 | 1, 2, 3, 4, 5, 7, 9, 27 | gsumcom3 19755 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘)))))) |
29 | 28 | eqcomd 2742 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∖ cdif 3907 〈cop 4592 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 supp csupp 8092 finSupp cfsupp 9305 Basecbs 17083 0gc0g 17321 Σg cgsu 17322 CMndccmn 19562 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-0g 17323 df-gsum 17324 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 |
This theorem is referenced by: (None) |
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