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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 19099 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋))) |
10 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
11 | 1, 2, 3, 4, 10, 6, 7, 8 | gsum2d2 19096 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋))))) |
12 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ∈ 𝑉) |
13 | 6 | ancom2s 648 | . . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴)) → 𝑋 ∈ 𝐵) |
14 | cnvfi 8808 | . . . 4 ⊢ (𝑈 ∈ Fin → ◡𝑈 ∈ Fin) | |
15 | 7, 14 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝑈 ∈ Fin) |
16 | ancom 463 | . . . . 5 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) | |
17 | vex 3499 | . . . . . . 7 ⊢ 𝑘 ∈ V | |
18 | vex 3499 | . . . . . . 7 ⊢ 𝑗 ∈ V | |
19 | 17, 18 | brcnv 5755 | . . . . . 6 ⊢ (𝑘◡𝑈𝑗 ↔ 𝑗𝑈𝑘) |
20 | 19 | notbii 322 | . . . . 5 ⊢ (¬ 𝑘◡𝑈𝑗 ↔ ¬ 𝑗𝑈𝑘) |
21 | 16, 20 | anbi12i 628 | . . . 4 ⊢ (((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗) ↔ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) |
22 | 21, 8 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ ((𝑘 ∈ 𝐶 ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑘◡𝑈𝑗)) → 𝑋 = 0 ) |
23 | 1, 2, 3, 5, 12, 13, 15, 22 | gsum2d2 19096 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶, 𝑗 ∈ 𝐴 ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ 𝑋))))) |
24 | 9, 11, 23 | 3eqtr3d 2866 | 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 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ↦ cmpt 5148 ◡ccnv 5556 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Fincfn 8511 Basecbs 16485 0gc0g 16715 Σg cgsu 16716 CMndccmn 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 |
This theorem is referenced by: gsumcom3fi 19101 gsumxp2 19102 |
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