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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 18031 | . . . 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 17197 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
7 | 3, 4, 6 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
8 | eqid 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
9 | tsms0.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
10 | 8, 5 | mndidcl 17131 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
11 | 3, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
12 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ (Base‘𝐺)) |
13 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 0 ) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
14 | 12, 13 | fmptd 6292 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ):𝐴⟶(Base‘𝐺)) |
15 | fconstmpt 5085 | . . . 4 ⊢ (𝐴 × { 0 }) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
16 | fvex 6113 | . . . . . . 7 ⊢ (0g‘𝐺) ∈ V | |
17 | 5, 16 | eqeltri 2684 | . . . . . 6 ⊢ 0 ∈ V |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
19 | 4, 18 | fczfsuppd 8176 | . . . 4 ⊢ (𝜑 → (𝐴 × { 0 }) finSupp 0 ) |
20 | 15, 19 | syl5eqbrr 4619 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ) finSupp 0 ) |
21 | 8, 5, 1, 9, 4, 14, 20 | tsmsid 21753 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
22 | 7, 21 | eqeltrrd 2689 | 1 ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 Basecbs 15695 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 CMndccmn 18016 TopSpctps 20519 tsums ctsu 21739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cntz 17573 df-cmn 18018 df-fbas 19564 df-fg 19565 df-top 20521 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tsms 21740 |
This theorem is referenced by: (None) |
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