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Mirrors > Home > MPE Home > Th. List > gsumcom3 | Structured version Visualization version GIF version |
Description: A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.) |
Ref | Expression |
---|---|
gsumcom3.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumcom3.z | ⊢ 0 = (0g‘𝐺) |
gsumcom3.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumcom3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumcom3.r | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
gsumcom3.f | ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵) |
gsumcom3.u | ⊢ (𝜑 → 𝑈 ∈ Fin) |
gsumcom3.n | ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 ) |
Ref | Expression |
---|---|
gsumcom3 | ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumcom3.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumcom3.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumcom3.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumcom3.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | gsumcom3.r | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
6 | gsumcom3.f | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵) | |
7 | gsumcom3.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ Fin) | |
8 | gsumcom3.n | . . 3 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | gsumcom 19745 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋))) |
10 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
11 | 1, 2, 3, 4, 10, 6, 7, 8 | gsum2d2 19742 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))))) |
12 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ∈ 𝑉) |
13 | 6 | ancom2s 648 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴)) → 𝑋 ∈ 𝐵) |
14 | cnvfi 9120 | . . . 4 ⊢ (𝑈 ∈ Fin → ◡𝑈 ∈ Fin) | |
15 | 7, 14 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝑈 ∈ Fin) |
16 | ancom 461 | . . . . 5 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) | |
17 | vex 3447 | . . . . . . 7 ⊢ 𝑘 ∈ V | |
18 | vex 3447 | . . . . . . 7 ⊢ 𝑗 ∈ V | |
19 | 17, 18 | brcnv 5836 | . . . . . 6 ⊢ (𝑘◡𝑈𝑗 ↔ 𝑗𝑈𝑘) |
20 | 19 | notbii 319 | . . . . 5 ⊢ (¬ 𝑘◡𝑈𝑗 ↔ ¬ 𝑗𝑈𝑘) |
21 | 16, 20 | anbi12i 627 | . . . 4 ⊢ (((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗) ↔ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) |
22 | 21, 8 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ ((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗)) → 𝑋 = 0 ) |
23 | 1, 2, 3, 5, 12, 13, 15, 22 | gsum2d2 19742 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋))))) |
24 | 9, 11, 23 | 3eqtr3d 2784 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)))) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ↦ cmpt 5186 ◡ccnv 5630 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 Fincfn 8879 Basecbs 17075 0gc0g 17313 Σg cgsu 17314 CMndccmn 19553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-0g 17315 df-gsum 17316 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-mulg 18864 df-cntz 19088 df-cmn 19555 |
This theorem is referenced by: gsumcom3fi 19747 gsumxp2 19748 |
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