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Mirrors > Home > MPE Home > Th. List > gsumsnfd | Structured version Visualization version GIF version |
Description: Group sum of a singleton, deduction form, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Thierry Arnoux, 28-Mar-2018.) (Revised by AV, 11-Dec-2019.) |
Ref | Expression |
---|---|
gsumsnd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumsnd.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsumsnd.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
gsumsnd.c | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
gsumsnd.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) |
gsumsnfd.p | ⊢ Ⅎ𝑘𝜑 |
gsumsnfd.c | ⊢ Ⅎ𝑘𝐶 |
Ref | Expression |
---|---|
gsumsnfd | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumsnfd.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
2 | elsni 4644 | . . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
3 | gsumsnd.s | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) | |
4 | 2, 3 | sylan2 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝐴 = 𝐶) |
5 | 1, 4 | mpteq2da 5245 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = (𝑘 ∈ {𝑀} ↦ 𝐶)) |
6 | 5 | oveq2d 7427 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶))) |
7 | gsumsnd.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
8 | snfi 9046 | . . . . 5 ⊢ {𝑀} ∈ Fin | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑀} ∈ Fin) |
10 | gsumsnd.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
11 | gsumsnfd.c | . . . . 5 ⊢ Ⅎ𝑘𝐶 | |
12 | gsumsnd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
13 | eqid 2730 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | 11, 12, 13 | gsumconstf 19844 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ {𝑀} ∈ Fin ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
15 | 7, 9, 10, 14 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
16 | 6, 15 | eqtrd 2770 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
17 | gsumsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
18 | hashsng 14333 | . . . 4 ⊢ (𝑀 ∈ 𝑉 → (♯‘{𝑀}) = 1) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘{𝑀}) = 1) |
20 | 19 | oveq1d 7426 | . 2 ⊢ (𝜑 → ((♯‘{𝑀})(.g‘𝐺)𝐶) = (1(.g‘𝐺)𝐶)) |
21 | 12, 13 | mulg1 18997 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (1(.g‘𝐺)𝐶) = 𝐶) |
22 | 10, 21 | syl 17 | . 2 ⊢ (𝜑 → (1(.g‘𝐺)𝐶) = 𝐶) |
23 | 16, 20, 22 | 3eqtrd 2774 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 Ⅎwnf 1783 ∈ wcel 2104 Ⅎwnfc 2881 {csn 4627 ↦ cmpt 5230 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 1c1 11113 ♯chash 14294 Basecbs 17148 Σg cgsu 17390 Mndcmnd 18659 .gcmg 18986 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-0g 17391 df-gsum 17392 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mulg 18987 df-cntz 19222 |
This theorem is referenced by: gsumsnd 19861 gsumsnf 19862 gsumunsnfd 19866 esumsnf 33360 gsumdifsndf 46857 |
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