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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 19584 | . . . 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 18651 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
7 | 3, 4, 6 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
8 | eqid 2733 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
9 | tsms0.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
10 | 8, 5 | mndidcl 18576 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
11 | 3, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
12 | 11 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ (Base‘𝐺)) |
13 | 12 | fmpttd 7064 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ):𝐴⟶(Base‘𝐺)) |
14 | fconstmpt 5695 | . . . 4 ⊢ (𝐴 × { 0 }) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
15 | 5 | fvexi 6857 | . . . . . 6 ⊢ 0 ∈ V |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
17 | 4, 16 | fczfsuppd 9328 | . . . 4 ⊢ (𝜑 → (𝐴 × { 0 }) finSupp 0 ) |
18 | 14, 17 | eqbrtrrid 5142 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ) finSupp 0 ) |
19 | 8, 5, 1, 9, 4, 13, 18 | tsmsid 23507 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
20 | 7, 19 | eqeltrrd 2835 | 1 ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 ↦ cmpt 5189 × cxp 5632 ‘cfv 6497 (class class class)co 7358 finSupp cfsupp 9308 Basecbs 17088 0gc0g 17326 Σg cgsu 17327 Mndcmnd 18561 CMndccmn 19567 TopSpctps 22297 tsums ctsu 23493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-0g 17328 df-gsum 17329 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-cntz 19102 df-cmn 19569 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 |
This theorem is referenced by: (None) |
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