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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 19026 | . 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 19023 | . 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 646 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴)) → 𝑋 ∈ 𝐵) |
14 | cnvfi 8794 | . . . 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 3495 | . . . . . . 7 ⊢ 𝑘 ∈ V | |
18 | vex 3495 | . . . . . . 7 ⊢ 𝑗 ∈ V | |
19 | 17, 18 | brcnv 5746 | . . . . . 6 ⊢ (𝑘◡𝑈𝑗 ↔ 𝑗𝑈𝑘) |
20 | 19 | notbii 321 | . . . . 5 ⊢ (¬ 𝑘◡𝑈𝑗 ↔ ¬ 𝑗𝑈𝑘) |
21 | 16, 20 | anbi12i 626 | . . . 4 ⊢ (((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗) ↔ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) |
22 | 21, 8 | sylan2b 593 | . . 3 ⊢ ((𝜑 ∧ ((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗)) → 𝑋 = 0 ) |
23 | 1, 2, 3, 5, 12, 13, 15, 22 | gsum2d2 19023 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋))))) |
24 | 9, 11, 23 | 3eqtr3d 2861 | 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 1528 ∈ wcel 2105 class class class wbr 5057 ↦ cmpt 5137 ◡ccnv 5547 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Fincfn 8497 Basecbs 16471 0gc0g 16701 Σg cgsu 16702 CMndccmn 18835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 |
This theorem is referenced by: gsumcom3fi 19028 gsumxp2 19029 |
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