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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 7454 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
7 | relxp 5537 | . . . 4 ⊢ Rel (𝐴 × 𝐶) | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → Rel (𝐴 × 𝐶)) |
9 | dmxpss 5995 | . . . 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 19085 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
14 | df-ima 5532 | . . . . . . 7 ⊢ ((𝐴 × 𝐶) “ {𝑗}) = ran ((𝐴 × 𝐶) ↾ {𝑗}) | |
15 | df-res 5531 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 × 𝐶) ∩ ({𝑗} × V)) | |
16 | inxp 5667 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ∩ ({𝑗} × V)) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) | |
17 | 15, 16 | eqtri 2821 | . . . . . . . . . 10 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) |
18 | simpr 488 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
19 | 18 | snssd 4702 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ⊆ 𝐴) |
20 | sseqin2 4142 | . . . . . . . . . . . 12 ⊢ ({𝑗} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑗}) = {𝑗}) | |
21 | 19, 20 | sylib 221 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐴 ∩ {𝑗}) = {𝑗}) |
22 | inv1 4302 | . . . . . . . . . . . 12 ⊢ (𝐶 ∩ V) = 𝐶 | |
23 | 22 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 ∩ V) = 𝐶) |
24 | 21, 23 | xpeq12d 5550 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) = ({𝑗} × 𝐶)) |
25 | 17, 24 | syl5eq 2845 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) ↾ {𝑗}) = ({𝑗} × 𝐶)) |
26 | 25 | rneqd 5772 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = ran ({𝑗} × 𝐶)) |
27 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑗 ∈ V | |
28 | 27 | snnz 4672 | . . . . . . . . 9 ⊢ {𝑗} ≠ ∅ |
29 | rnxp 5994 | . . . . . . . . 9 ⊢ ({𝑗} ≠ ∅ → ran ({𝑗} × 𝐶) = 𝐶) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ ran ({𝑗} × 𝐶) = 𝐶 |
31 | 26, 30 | eqtrdi 2849 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = 𝐶) |
32 | 14, 31 | syl5eq 2845 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) “ {𝑗}) = 𝐶) |
33 | 32 | mpteq1d 5119 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))) |
34 | 33 | oveq2d 7151 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) |
35 | 34 | mpteq2dva 5125 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) |
36 | 35 | oveq2d 7151 | . 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 2833 | 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 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 Rel wrel 5524 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 finSupp cfsupp 8817 Basecbs 16475 0gc0g 16705 Σg cgsu 16706 CMndccmn 18898 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: tsmsxplem1 22758 tsmsxplem2 22759 fedgmullem1 31113 fedgmullem2 31114 |
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