![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7770 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
7 | relxp 5707 | . . . 4 ⊢ Rel (𝐴 × 𝐶) | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → Rel (𝐴 × 𝐶)) |
9 | dmxpss 6193 | . . . 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 20005 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
14 | df-ima 5702 | . . . . . . 7 ⊢ ((𝐴 × 𝐶) “ {𝑗}) = ran ((𝐴 × 𝐶) ↾ {𝑗}) | |
15 | df-res 5701 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 × 𝐶) ∩ ({𝑗} × V)) | |
16 | inxp 5845 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ∩ ({𝑗} × V)) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) | |
17 | 15, 16 | eqtri 2763 | . . . . . . . . . 10 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) |
18 | simpr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
19 | 18 | snssd 4814 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ⊆ 𝐴) |
20 | sseqin2 4231 | . . . . . . . . . . . 12 ⊢ ({𝑗} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑗}) = {𝑗}) | |
21 | 19, 20 | sylib 218 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐴 ∩ {𝑗}) = {𝑗}) |
22 | inv1 4404 | . . . . . . . . . . . 12 ⊢ (𝐶 ∩ V) = 𝐶 | |
23 | 22 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 ∩ V) = 𝐶) |
24 | 21, 23 | xpeq12d 5720 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) = ({𝑗} × 𝐶)) |
25 | 17, 24 | eqtrid 2787 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) ↾ {𝑗}) = ({𝑗} × 𝐶)) |
26 | 25 | rneqd 5952 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = ran ({𝑗} × 𝐶)) |
27 | vex 3482 | . . . . . . . . . 10 ⊢ 𝑗 ∈ V | |
28 | 27 | snnz 4781 | . . . . . . . . 9 ⊢ {𝑗} ≠ ∅ |
29 | rnxp 6192 | . . . . . . . . 9 ⊢ ({𝑗} ≠ ∅ → ran ({𝑗} × 𝐶) = 𝐶) | |
30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ ran ({𝑗} × 𝐶) = 𝐶 |
31 | 26, 30 | eqtrdi 2791 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = 𝐶) |
32 | 14, 31 | eqtrid 2787 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) “ {𝑗}) = 𝐶) |
33 | 32 | mpteq1d 5243 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))) |
34 | 33 | oveq2d 7447 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))) |
35 | 34 | mpteq2dva 5248 | . . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) |
36 | 35 | oveq2d 7447 | . 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 2775 | 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Rel wrel 5694 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 finSupp cfsupp 9399 Basecbs 17245 0gc0g 17486 Σg cgsu 17487 CMndccmn 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 |
This theorem is referenced by: tsmsxplem1 24177 tsmsxplem2 24178 fedgmullem1 33657 fedgmullem2 33658 evlselv 42574 |
Copyright terms: Public domain | W3C validator |