| 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 7771 | . . 3 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
| 7 | relxp 5703 | . . . 4 ⊢ Rel (𝐴 × 𝐶) | |
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → Rel (𝐴 × 𝐶)) |
| 9 | dmxpss 6191 | . . . 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 19990 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ ((𝐴 × 𝐶) “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
| 14 | df-ima 5698 | . . . . . . 7 ⊢ ((𝐴 × 𝐶) “ {𝑗}) = ran ((𝐴 × 𝐶) ↾ {𝑗}) | |
| 15 | df-res 5697 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 × 𝐶) ∩ ({𝑗} × V)) | |
| 16 | inxp 5842 | . . . . . . . . . . 11 ⊢ ((𝐴 × 𝐶) ∩ ({𝑗} × V)) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) | |
| 17 | 15, 16 | eqtri 2765 | . . . . . . . . . 10 ⊢ ((𝐴 × 𝐶) ↾ {𝑗}) = ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) |
| 18 | simpr 484 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
| 19 | 18 | snssd 4809 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → {𝑗} ⊆ 𝐴) |
| 20 | sseqin2 4223 | . . . . . . . . . . . 12 ⊢ ({𝑗} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑗}) = {𝑗}) | |
| 21 | 19, 20 | sylib 218 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐴 ∩ {𝑗}) = {𝑗}) |
| 22 | inv1 4398 | . . . . . . . . . . . 12 ⊢ (𝐶 ∩ V) = 𝐶 | |
| 23 | 22 | a1i 11 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐶 ∩ V) = 𝐶) |
| 24 | 21, 23 | xpeq12d 5716 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 ∩ {𝑗}) × (𝐶 ∩ V)) = ({𝑗} × 𝐶)) |
| 25 | 17, 24 | eqtrid 2789 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) ↾ {𝑗}) = ({𝑗} × 𝐶)) |
| 26 | 25 | rneqd 5949 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = ran ({𝑗} × 𝐶)) |
| 27 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑗 ∈ V | |
| 28 | 27 | snnz 4776 | . . . . . . . . 9 ⊢ {𝑗} ≠ ∅ |
| 29 | rnxp 6190 | . . . . . . . . 9 ⊢ ({𝑗} ≠ ∅ → ran ({𝑗} × 𝐶) = 𝐶) | |
| 30 | 28, 29 | ax-mp 5 | . . . . . . . 8 ⊢ ran ({𝑗} × 𝐶) = 𝐶 |
| 31 | 26, 30 | eqtrdi 2793 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ran ((𝐴 × 𝐶) ↾ {𝑗}) = 𝐶) |
| 32 | 14, 31 | eqtrid 2789 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐴 × 𝐶) “ {𝑗}) = 𝐶) |
| 33 | 32 | mpteq1d 5237 | . . . . 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 5242 | . . 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 2777 | 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Rel wrel 5690 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 finSupp cfsupp 9401 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: tsmsxplem1 24161 tsmsxplem2 24162 fedgmullem1 33680 fedgmullem2 33681 evlselv 42597 |
| Copyright terms: Public domain | W3C validator |