![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > gsumbagdiag | Structured version Visualization version GIF version |
Description: Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 15810 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.) |
Ref | Expression |
---|---|
gsumbagdiag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
gsumbagdiag.s | ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} |
gsumbagdiag.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
gsumbagdiag.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumbagdiag.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumbagdiag.x | ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
gsumbagdiag | ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝑆, 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝑆, 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)} ↦ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumbagdiag.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2735 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsumbagdiag.g | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumbagdiag.s | . . 3 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} | |
5 | gsumbagdiag.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
6 | gsumbagdiag.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | 6 | psrbaglefi 21964 | . . . 4 ⊢ (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin) |
8 | 5, 7 | syl 17 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin) |
9 | 4, 8 | eqeltrid 2843 | . 2 ⊢ (𝜑 → 𝑆 ∈ Fin) |
10 | ovex 7464 | . . . 4 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
11 | 6, 10 | rab2ex 5348 | . . 3 ⊢ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ∈ V |
12 | 11 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑆) → {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ∈ V) |
13 | gsumbagdiag.x | . 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵) | |
14 | xpfi 9356 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ 𝑆 ∈ Fin) → (𝑆 × 𝑆) ∈ Fin) | |
15 | 9, 9, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ Fin) |
16 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑗 ∈ 𝑆) | |
17 | 6, 4, 5 | gsumbagdiaglem 21968 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → (𝑘 ∈ 𝑆 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)})) |
18 | 17 | simpld 494 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑘 ∈ 𝑆) |
19 | brxp 5738 | . . . . 5 ⊢ (𝑗(𝑆 × 𝑆)𝑘 ↔ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆)) | |
20 | 16, 18, 19 | sylanbrc 583 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑗(𝑆 × 𝑆)𝑘) |
21 | 20 | pm2.24d 151 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → (¬ 𝑗(𝑆 × 𝑆)𝑘 → 𝑋 = (0g‘𝐺))) |
22 | 21 | impr 454 | . 2 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)}) ∧ ¬ 𝑗(𝑆 × 𝑆)𝑘)) → 𝑋 = (0g‘𝐺)) |
23 | 6, 4, 5 | gsumbagdiaglem 21968 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)})) → (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) |
24 | 17, 23 | impbida 801 | . 2 ⊢ (𝜑 → ((𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)}) ↔ (𝑘 ∈ 𝑆 ∧ 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)}))) |
25 | 1, 2, 3, 9, 12, 13, 15, 22, 9, 24 | gsumcom2 20008 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝑆, 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝑆, 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)} ↦ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 class class class wbr 5148 × cxp 5687 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ∘f cof 7695 ∘r cofr 7696 ↑m cmap 8865 Fincfn 8984 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 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-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-ofr 7698 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-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 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-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cntz 19348 df-cmn 19815 |
This theorem is referenced by: psrass1lem 21970 |
Copyright terms: Public domain | W3C validator |