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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 7474 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
7 | relxp 5573 | . . . 4 ⊢ Rel (𝐴 × 𝐶) | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → Rel (𝐴 × 𝐶)) |
9 | dmxpss 6028 | . . . 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 19092 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
14 | df-ima 5568 | . . . . . . 7 ⊢ ((𝐴 × 𝐶) “ {𝑗}) = ran ((𝐴 × 𝐶) ↾ {𝑗}) | |
15 | df-res 5567 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 × 𝐶) ∩ ({𝑗} × V)) | |
16 | inxp 5703 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ∩ ({𝑗} × V)) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) | |
17 | 15, 16 | eqtri 2844 | . . . . . . . . . 10 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) |
18 | simpr 487 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
19 | 18 | snssd 4742 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ⊆ 𝐴) |
20 | sseqin2 4192 | . . . . . . . . . . . 12 ⊢ ({𝑗} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑗}) = {𝑗}) | |
21 | 19, 20 | sylib 220 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐴 ∩ {𝑗}) = {𝑗}) |
22 | inv1 4348 | . . . . . . . . . . . 12 ⊢ (𝐶 ∩ V) = 𝐶 | |
23 | 22 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 ∩ V) = 𝐶) |
24 | 21, 23 | xpeq12d 5586 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) = ({𝑗} × 𝐶)) |
25 | 17, 24 | syl5eq 2868 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) ↾ {𝑗}) = ({𝑗} × 𝐶)) |
26 | 25 | rneqd 5808 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = ran ({𝑗} × 𝐶)) |
27 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑗 ∈ V | |
28 | 27 | snnz 4711 | . . . . . . . . 9 ⊢ {𝑗} ≠ ∅ |
29 | rnxp 6027 | . . . . . . . . 9 ⊢ ({𝑗} ≠ ∅ → ran ({𝑗} × 𝐶) = 𝐶) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ ran ({𝑗} × 𝐶) = 𝐶 |
31 | 26, 30 | syl6eq 2872 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = 𝐶) |
32 | 14, 31 | syl5eq 2868 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) “ {𝑗}) = 𝐶) |
33 | 32 | mpteq1d 5155 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))) |
34 | 33 | oveq2d 7172 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) |
35 | 34 | mpteq2dva 5161 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) |
36 | 35 | oveq2d 7172 | . 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 2856 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 Rel wrel 5560 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 finSupp cfsupp 8833 Basecbs 16483 0gc0g 16713 Σg cgsu 16714 CMndccmn 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: tsmsxplem1 22761 tsmsxplem2 22762 fedgmullem1 31025 fedgmullem2 31026 |
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