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| Mirrors > Home > MPE Home > Th. List > gsumxp2 | Structured version Visualization version GIF version | ||
| Description: Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.) |
| Ref | Expression |
|---|---|
| gsumxp2.b | ⊢ 𝐵 = (Base‘𝐺) |
| gsumxp2.z | ⊢ 0 = (0g‘𝐺) |
| gsumxp2.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| gsumxp2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| gsumxp2.r | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
| gsumxp2.f | ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) |
| gsumxp2.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
| Ref | Expression |
|---|---|
| gsumxp2 | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | gsumxp2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | gsumxp2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 3 | gsumxp2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 4 | gsumxp2.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 5 | gsumxp2.r | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
| 6 | gsumxp2.f | . . . 4 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵) | |
| 7 | 6 | fovcdmda 7604 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (𝑗𝐹𝑘) ∈ 𝐵) |
| 8 | gsumxp2.w | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
| 9 | 8 | fsuppimpd 9409 | . . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
| 10 | simpl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝜑) | |
| 11 | opelxpi 5722 | . . . . . . . . 9 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) → 〈𝑗, 𝑘〉 ∈ (𝐴 × 𝐶)) | |
| 12 | 11 | ad2antlr 727 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 × 𝐶)) |
| 13 | simpr 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
| 14 | 12, 13 | eldifd 3962 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ ((𝐴 × 𝐶) ∖ (𝐹 supp 0 ))) |
| 15 | ssidd 4007 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) | |
| 16 | 4, 5 | xpexd 7771 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 × 𝐶) ∈ V) |
| 17 | 2 | fvexi 6920 | . . . . . . . . 9 ⊢ 0 ∈ V |
| 18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ V) |
| 19 | 6, 15, 16, 18 | suppssr 8220 | . . . . . . 7 ⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ ((𝐴 × 𝐶) ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
| 20 | 10, 14, 19 | syl2an2r 685 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
| 21 | 20 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ) → (𝐹‘〈𝑗, 𝑘〉) = 0 )) |
| 22 | df-br 5144 | . . . . . 6 ⊢ (𝑗(𝐹 supp 0 )𝑘 ↔ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) | |
| 23 | 22 | notbii 320 | . . . . 5 ⊢ (¬ 𝑗(𝐹 supp 0 )𝑘 ↔ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) |
| 24 | df-ov 7434 | . . . . . 6 ⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) | |
| 25 | 24 | eqeq1i 2742 | . . . . 5 ⊢ ((𝑗𝐹𝑘) = 0 ↔ (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
| 26 | 21, 23, 25 | 3imtr4g 296 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → (¬ 𝑗(𝐹 supp 0 )𝑘 → (𝑗𝐹𝑘) = 0 )) |
| 27 | 26 | impr 454 | . . 3 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗(𝐹 supp 0 )𝑘)) → (𝑗𝐹𝑘) = 0 ) |
| 28 | 1, 2, 3, 4, 5, 7, 9, 27 | gsumcom3 19996 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘)))))) |
| 29 | 28 | eqcomd 2743 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 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: (None) |
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