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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 18922 | . . . 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 18000 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
7 | 3, 4, 6 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
8 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
9 | tsms0.2 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TopSp) | |
10 | 8, 5 | mndidcl 17926 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺)) |
11 | 3, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ (Base‘𝐺)) |
13 | 12 | fmpttd 6879 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ):𝐴⟶(Base‘𝐺)) |
14 | fconstmpt 5614 | . . . 4 ⊢ (𝐴 × { 0 }) = (𝑥 ∈ 𝐴 ↦ 0 ) | |
15 | 5 | fvexi 6684 | . . . . . 6 ⊢ 0 ∈ V |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
17 | 4, 16 | fczfsuppd 8851 | . . . 4 ⊢ (𝜑 → (𝐴 × { 0 }) finSupp 0 ) |
18 | 14, 17 | eqbrtrrid 5102 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 0 ) finSupp 0 ) |
19 | 8, 5, 1, 9, 4, 13, 18 | tsmsid 22748 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
20 | 7, 19 | eqeltrrd 2914 | 1 ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 (class class class)co 7156 finSupp cfsupp 8833 Basecbs 16483 0gc0g 16713 Σg cgsu 16714 Mndcmnd 17911 CMndccmn 18906 TopSpctps 21540 tsums ctsu 22734 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-0g 16715 df-gsum 16716 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-cntz 18447 df-cmn 18908 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-topsp 21541 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 |
This theorem is referenced by: (None) |
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