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Mirrors > Home > MPE Home > Th. List > tsms0 | Structured version Visualization version GIF version |
Description: The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.) |
Ref | Expression |
---|---|
tsms0.z | ⊢ 0 = (0g‘𝐺) |
tsms0.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
tsms0.2 | ⊢ (𝜑 → 𝐺 ∈ TopSp) |
tsms0.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
tsms0 | ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tsms0.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
2 | cmnmnd 18297 | . . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
4 | tsms0.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | tsms0.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
6 | 5 | gsumz 17464 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
7 | 3, 4, 6 | syl2anc 696 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
8 | eqid 2692 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
9 | tsms0.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
10 | 8, 5 | mndidcl 17398 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
11 | 3, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
12 | 11 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ (Base‘𝐺)) |
13 | eqid 2692 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 0 ) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
14 | 12, 13 | fmptd 6468 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ):𝐴⟶(Base‘𝐺)) |
15 | fconstmpt 5240 | . . . 4 ⊢ (𝐴 × { 0 }) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
16 | fvex 6282 | . . . . . . 7 ⊢ (0g‘𝐺) ∈ V | |
17 | 5, 16 | eqeltri 2767 | . . . . . 6 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
19 | 4, 18 | fczfsuppd 8377 | . . . 4 ⊢ (𝜑 → (𝐴 × { 0 }) finSupp 0 ) |
20 | 15, 19 | syl5eqbrr 4764 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ) finSupp 0 ) |
21 | 8, 5, 1, 9, 4, 14, 20 | tsmsid 22033 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
22 | 7, 21 | eqeltrrd 2772 | 1 ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1564 ∈ wcel 2071 Vcvv 3272 {csn 4253 ↦ cmpt 4805 × cxp 5184 ‘cfv 5969 (class class class)co 6733 finSupp cfsupp 8359 Basecbs 15948 0gc0g 16191 Σg cgsu 16192 Mndcmnd 17384 CMndccmn 18282 TopSpctps 20827 tsums ctsu 22019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-rep 4847 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-int 4552 df-iun 4598 df-iin 4599 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-se 5146 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-isom 5978 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-supp 7384 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-1o 7648 df-oadd 7652 df-er 7830 df-map 7944 df-en 8041 df-dom 8042 df-sdom 8043 df-fin 8044 df-fsupp 8360 df-oi 8499 df-card 8846 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-nn 11102 df-n0 11374 df-z 11459 df-uz 11769 df-fz 12409 df-fzo 12549 df-seq 12885 df-hash 13201 df-0g 16193 df-gsum 16194 df-mgm 17332 df-sgrp 17374 df-mnd 17385 df-cntz 17839 df-cmn 18284 df-fbas 19834 df-fg 19835 df-top 20790 df-topon 20807 df-topsp 20828 df-cld 20914 df-ntr 20915 df-cls 20916 df-nei 20993 df-fil 21740 df-fm 21832 df-flim 21833 df-flf 21834 df-tsms 22020 |
This theorem is referenced by: (None) |
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