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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 7510 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (𝑗𝐹𝑘) ∈ 𝐵) |
8 | gsumxp2.w | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
9 | 8 | fsuppimpd 9238 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
10 | simpl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝜑) | |
11 | opelxpi 5662 | . . . . . . . . 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 486 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
14 | 12, 13 | eldifd 3913 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ ((𝐴 × 𝐶) ∖ (𝐹 supp 0 ))) |
15 | ssidd 3959 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
16 | 4, 5 | xpexd 7668 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
17 | 2 | fvexi 6844 | . . . . . . . . 9 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ V) |
19 | 6, 15, 16, 18 | suppssr 8087 | . . . . . . 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 414 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ) → (𝐹‘〈𝑗, 𝑘〉) = 0 )) |
22 | df-br 5098 | . . . . . 6 ⊢ (𝑗(𝐹 supp 0 )𝑘 ↔ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
23 | 22 | notbii 320 | . . . . 5 ⊢ (¬ 𝑗(𝐹 supp 0 )𝑘 ↔ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) |
24 | df-ov 7345 | . . . . . 6 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
25 | 24 | eqeq1i 2742 | . . . . 5 ⊢ ((𝑗𝐹𝑘) = 0 ↔ (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
26 | 21, 23, 25 | 3imtr4g 296 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 𝑗(𝐹 supp 0 )𝑘 → (𝑗𝐹𝑘) = 0 )) |
27 | 26 | impr 456 | . . 3 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗(𝐹 supp 0 )𝑘)) → (𝑗𝐹𝑘) = 0 ) |
28 | 1, 2, 3, 4, 5, 7, 9, 27 | gsumcom3 19674 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘)))))) |
29 | 28 | eqcomd 2743 | 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 397 = wceq 1541 ∈ wcel 2106 Vcvv 3442 ∖ cdif 3899 〈cop 4584 class class class wbr 5097 ↦ cmpt 5180 × cxp 5623 ⟶wf 6480 ‘cfv 6484 (class class class)co 7342 supp csupp 8052 finSupp cfsupp 9231 Basecbs 17010 0gc0g 17248 Σg cgsu 17249 CMndccmn 19482 |
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 2708 ax-rep 5234 ax-sep 5248 ax-nul 5255 ax-pow 5313 ax-pr 5377 ax-un 7655 ax-cnex 11033 ax-resscn 11034 ax-1cn 11035 ax-icn 11036 ax-addcl 11037 ax-addrcl 11038 ax-mulcl 11039 ax-mulrcl 11040 ax-mulcom 11041 ax-addass 11042 ax-mulass 11043 ax-distr 11044 ax-i2m1 11045 ax-1ne0 11046 ax-1rid 11047 ax-rnegex 11048 ax-rrecex 11049 ax-cnre 11050 ax-pre-lttri 11051 ax-pre-lttrn 11052 ax-pre-ltadd 11053 ax-pre-mulgt0 11054 |
This theorem depends on definitions: df-bi 206 df-an 398 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3732 df-csb 3848 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3921 df-nul 4275 df-if 4479 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4858 df-int 4900 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5181 df-tr 5215 df-id 5523 df-eprel 5529 df-po 5537 df-so 5538 df-fr 5580 df-se 5581 df-we 5582 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6243 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6436 df-fun 6486 df-fn 6487 df-f 6488 df-f1 6489 df-fo 6490 df-f1o 6491 df-fv 6492 df-isom 6493 df-riota 7298 df-ov 7345 df-oprab 7346 df-mpo 7347 df-of 7600 df-om 7786 df-1st 7904 df-2nd 7905 df-supp 8053 df-frecs 8172 df-wrecs 8203 df-recs 8277 df-rdg 8316 df-1o 8372 df-er 8574 df-en 8810 df-dom 8811 df-sdom 8812 df-fin 8813 df-fsupp 9232 df-oi 9372 df-card 9801 df-pnf 11117 df-mnf 11118 df-xr 11119 df-ltxr 11120 df-le 11121 df-sub 11313 df-neg 11314 df-nn 12080 df-2 12142 df-n0 12340 df-z 12426 df-uz 12689 df-fz 13346 df-fzo 13489 df-seq 13828 df-hash 14151 df-sets 16963 df-slot 16981 df-ndx 16993 df-base 17011 df-ress 17040 df-plusg 17073 df-0g 17250 df-gsum 17251 df-mre 17393 df-mrc 17394 df-acs 17396 df-mgm 18424 df-sgrp 18473 df-mnd 18484 df-submnd 18529 df-mulg 18798 df-cntz 19020 df-cmn 19484 |
This theorem is referenced by: (None) |
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