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Mirrors > Home > MPE Home > Th. List > gsumxp | Structured version Visualization version GIF version |
Description: Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
gsumxp.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumxp.z | ⊢ 0 = (0g‘𝐺) |
gsumxp.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumxp.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumxp.r | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
gsumxp.f | ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) |
gsumxp.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
gsumxp | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumxp.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumxp.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumxp.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumxp.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | gsumxp.r | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
6 | 4, 5 | xpexd 7738 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
7 | relxp 5695 | . . . 4 ⊢ Rel (𝐴 × 𝐶) | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → Rel (𝐴 × 𝐶)) |
9 | dmxpss 6171 | . . . 4 ⊢ dom (𝐴 × 𝐶) ⊆ 𝐴 | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → dom (𝐴 × 𝐶) ⊆ 𝐴) |
11 | gsumxp.f | . . 3 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) | |
12 | gsumxp.w | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
13 | 1, 2, 3, 6, 8, 4, 10, 11, 12 | gsum2d 19840 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
14 | df-ima 5690 | . . . . . . 7 ⊢ ((𝐴 × 𝐶) “ {𝑗}) = ran ((𝐴 × 𝐶) ↾ {𝑗}) | |
15 | df-res 5689 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 × 𝐶) ∩ ({𝑗} × V)) | |
16 | inxp 5833 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ∩ ({𝑗} × V)) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) | |
17 | 15, 16 | eqtri 2761 | . . . . . . . . . 10 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) |
18 | simpr 486 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
19 | 18 | snssd 4813 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ⊆ 𝐴) |
20 | sseqin2 4216 | . . . . . . . . . . . 12 ⊢ ({𝑗} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑗}) = {𝑗}) | |
21 | 19, 20 | sylib 217 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐴 ∩ {𝑗}) = {𝑗}) |
22 | inv1 4395 | . . . . . . . . . . . 12 ⊢ (𝐶 ∩ V) = 𝐶 | |
23 | 22 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 ∩ V) = 𝐶) |
24 | 21, 23 | xpeq12d 5708 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) = ({𝑗} × 𝐶)) |
25 | 17, 24 | eqtrid 2785 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) ↾ {𝑗}) = ({𝑗} × 𝐶)) |
26 | 25 | rneqd 5938 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = ran ({𝑗} × 𝐶)) |
27 | vex 3479 | . . . . . . . . . 10 ⊢ 𝑗 ∈ V | |
28 | 27 | snnz 4781 | . . . . . . . . 9 ⊢ {𝑗} ≠ ∅ |
29 | rnxp 6170 | . . . . . . . . 9 ⊢ ({𝑗} ≠ ∅ → ran ({𝑗} × 𝐶) = 𝐶) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ ran ({𝑗} × 𝐶) = 𝐶 |
31 | 26, 30 | eqtrdi 2789 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = 𝐶) |
32 | 14, 31 | eqtrid 2785 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) “ {𝑗}) = 𝐶) |
33 | 32 | mpteq1d 5244 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))) |
34 | 33 | oveq2d 7425 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) |
35 | 34 | mpteq2dva 5249 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) |
36 | 35 | oveq2d 7425 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
37 | 13, 36 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 dom cdm 5677 ran crn 5678 ↾ cres 5679 “ cima 5680 Rel wrel 5682 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 finSupp cfsupp 9361 Basecbs 17144 0gc0g 17385 Σg cgsu 17386 CMndccmn 19648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-gsum 17388 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 |
This theorem is referenced by: tsmsxplem1 23657 tsmsxplem2 23658 fedgmullem1 32745 fedgmullem2 32746 evlselv 41207 |
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