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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 4586 | . . . . . 6 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
3 | gsumsnd.s | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) | |
4 | 2, 3 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝐴 = 𝐶) |
5 | 1, 4 | mpteq2da 5162 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = (𝑘 ∈ {𝑀} ↦ 𝐶)) |
6 | 5 | oveq2d 7174 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶))) |
7 | gsumsnd.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
8 | snfi 8596 | . . . . 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 2823 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
14 | 11, 12, 13 | gsumconstf 19057 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ {𝑀} ∈ Fin ∧ 𝐶 ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
15 | 7, 9, 10, 14 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐶)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
16 | 6, 15 | eqtrd 2858 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = ((♯‘{𝑀})(.g‘𝐺)𝐶)) |
17 | gsumsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
18 | hashsng 13733 | . . . 4 ⊢ (𝑀 ∈ 𝑉 → (♯‘{𝑀}) = 1) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘{𝑀}) = 1) |
20 | 19 | oveq1d 7173 | . 2 ⊢ (𝜑 → ((♯‘{𝑀})(.g‘𝐺)𝐶) = (1(.g‘𝐺)𝐶)) |
21 | 12, 13 | mulg1 18237 | . . 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 2862 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 {csn 4569 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 1c1 10540 ♯chash 13693 Basecbs 16485 Σg cgsu 16716 Mndcmnd 17913 .gcmg 18226 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mulg 18227 df-cntz 18449 |
This theorem is referenced by: gsumsnd 19074 gsumsnf 19075 gsumunsnfd 19079 esumsnf 31325 gsumdifsndf 44095 |
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