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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 4142 | . . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
3 | gsumsnd.s | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) | |
4 | 2, 3 | sylan2 490 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝐴 = 𝐶) |
5 | 1, 4 | mpteq2da 4671 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = (𝑘 ∈ {𝑀} ↦ 𝐶)) |
6 | 5 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶))) |
7 | gsumsnd.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
8 | snfi 7923 | . . . . 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 2610 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | 11, 12, 13 | gsumconstf 18158 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ {𝑀} ∈ Fin ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((#‘{𝑀})(.g‘𝐺)𝐶)) |
15 | 7, 9, 10, 14 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((#‘{𝑀})(.g‘𝐺)𝐶)) |
16 | 6, 15 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = ((#‘{𝑀})(.g‘𝐺)𝐶)) |
17 | gsumsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
18 | hashsng 13020 | . . . 4 ⊢ (𝑀 ∈ 𝑉 → (#‘{𝑀}) = 1) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (#‘{𝑀}) = 1) |
20 | 19 | oveq1d 6564 | . 2 ⊢ (𝜑 → ((#‘{𝑀})(.g‘𝐺)𝐶) = (1(.g‘𝐺)𝐶)) |
21 | 12, 13 | mulg1 17371 | . . 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 2648 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 #chash 12979 Basecbs 15695 Σg cgsu 15924 Mndcmnd 17117 .gcmg 17363 |
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-inf2 8421 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-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-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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-mulg 17364 df-cntz 17573 |
This theorem is referenced by: gsumsnd 18175 gsumsnf 18176 gsumunsnfd 18179 esumsnf 29453 gsumdifsndf 41937 |
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