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| Mirrors > Home > MPE Home > Th. List > tsmslem1 | Structured version Visualization version GIF version | ||
| Description: The finite partial sums of a function 𝐹 are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| tsmslem1.b | ⊢ 𝐵 = (Base‘𝐺) | 
| tsmslem1.s | ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) | 
| tsmslem1.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) | 
| tsmslem1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) | 
| tsmslem1.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| Ref | Expression | 
|---|---|
| tsmslem1 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺 Σg (𝐹 ↾ 𝑋)) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tsmslem1.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | eqid 2737 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 3 | tsmslem1.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 4 | 3 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝐺 ∈ CMnd) | 
| 5 | simpr 484 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
| 6 | tsmslem1.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 7 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝐹:𝐴⟶𝐵) | 
| 8 | tsmslem1.s | . . . . 5 ⊢ 𝑆 = (𝒫 𝐴 ∩ Fin) | |
| 9 | 5, 8 | eleqtrdi 2851 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (𝒫 𝐴 ∩ Fin)) | 
| 10 | elfpw 9394 | . . . . 5 ⊢ (𝑋 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin)) | |
| 11 | 10 | simplbi 497 | . . . 4 ⊢ (𝑋 ∈ (𝒫 𝐴 ∩ Fin) → 𝑋 ⊆ 𝐴) | 
| 12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) | 
| 13 | 7, 12 | fssresd 6775 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑋):𝑋⟶𝐵) | 
| 14 | 9 | elin2d 4205 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ Fin) | 
| 15 | fvexd 6921 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (0g‘𝐺) ∈ V) | |
| 16 | 13, 14, 15 | fdmfifsupp 9415 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹 ↾ 𝑋) finSupp (0g‘𝐺)) | 
| 17 | 1, 2, 4, 5, 13, 16 | gsumcl 19933 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺 Σg (𝐹 ↾ 𝑋)) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 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-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-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-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-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cntz 19335 df-cmn 19800 | 
| This theorem is referenced by: eltsms 24141 haustsms 24144 tsmscls 24146 tsmsmhm 24154 tsmsadd 24155 | 
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