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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 4542 | . . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
3 | gsumsnd.s | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) | |
4 | 2, 3 | sylan2 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝐴 = 𝐶) |
5 | 1, 4 | mpteq2da 5124 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = (𝑘 ∈ {𝑀} ↦ 𝐶)) |
6 | 5 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶))) |
7 | gsumsnd.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
8 | snfi 8577 | . . . . 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 2798 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | 11, 12, 13 | gsumconstf 19048 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ {𝑀} ∈ Fin ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
15 | 7, 9, 10, 14 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
16 | 6, 15 | eqtrd 2833 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
17 | gsumsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
18 | hashsng 13726 | . . . 4 ⊢ (𝑀 ∈ 𝑉 → (♯‘{𝑀}) = 1) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘{𝑀}) = 1) |
20 | 19 | oveq1d 7150 | . 2 ⊢ (𝜑 → ((♯‘{𝑀})(.g‘𝐺)𝐶) = (1(.g‘𝐺)𝐶)) |
21 | 12, 13 | mulg1 18227 | . . 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 2837 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 {csn 4525 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 1c1 10527 ♯chash 13686 Basecbs 16475 Σg cgsu 16706 Mndcmnd 17903 .gcmg 18216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 df-cntz 18439 |
This theorem is referenced by: gsumsnd 19065 gsumsnf 19066 gsumunsnfd 19070 esumsnf 31433 gsumdifsndf 44441 |
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