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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumtp | Structured version Visualization version GIF version |
Description: Group sum of an unordered triple. (Contributed by Thierry Arnoux, 22-Jun-2025.) |
Ref | Expression |
---|---|
gsumtp.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumtp.p | ⊢ + = (+g‘𝐺) |
gsumtp.s | ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) |
gsumtp.t | ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐷) |
gsumtp.u | ⊢ (𝑘 = 𝑂 → 𝐴 = 𝐸) |
gsumtp.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumtp.2 | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
gsumtp.3 | ⊢ (𝜑 → 𝑁 ∈ 𝑊) |
gsumtp.4 | ⊢ (𝜑 → 𝑂 ∈ 𝑋) |
gsumtp.5 | ⊢ (𝜑 → 𝑀 ≠ 𝑁) |
gsumtp.6 | ⊢ (𝜑 → 𝑁 ≠ 𝑂) |
gsumtp.7 | ⊢ (𝜑 → 𝑀 ≠ 𝑂) |
gsumtp.8 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
gsumtp.9 | ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
gsumtp.10 | ⊢ (𝜑 → 𝐸 ∈ 𝐵) |
Ref | Expression |
---|---|
gsumtp | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁, 𝑂} ↦ 𝐴)) = ((𝐶 + 𝐷) + 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumtp.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumtp.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | gsumtp.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | tpfi 9389 | . . . 4 ⊢ {𝑀, 𝑁, 𝑂} ∈ Fin | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑀, 𝑁, 𝑂} ∈ Fin) |
6 | gsumtp.s | . . . . . 6 ⊢ (𝑘 = 𝑀 → 𝐴 = 𝐶) | |
7 | 6 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑀) → 𝐴 = 𝐶) |
8 | gsumtp.8 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
9 | 8 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑀) → 𝐶 ∈ 𝐵) |
10 | 7, 9 | eqeltrd 2838 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑀) → 𝐴 ∈ 𝐵) |
11 | gsumtp.t | . . . . . 6 ⊢ (𝑘 = 𝑁 → 𝐴 = 𝐷) | |
12 | 11 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑁) → 𝐴 = 𝐷) |
13 | gsumtp.9 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝐵) | |
14 | 13 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑁) → 𝐷 ∈ 𝐵) |
15 | 12, 14 | eqeltrd 2838 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑁) → 𝐴 ∈ 𝐵) |
16 | gsumtp.u | . . . . . 6 ⊢ (𝑘 = 𝑂 → 𝐴 = 𝐸) | |
17 | 16 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑂) → 𝐴 = 𝐸) |
18 | gsumtp.10 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐵) | |
19 | 18 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑂) → 𝐸 ∈ 𝐵) |
20 | 17, 19 | eqeltrd 2838 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) ∧ 𝑘 = 𝑂) → 𝐴 ∈ 𝐵) |
21 | eltpi 4711 | . . . . 5 ⊢ (𝑘 ∈ {𝑀, 𝑁, 𝑂} → (𝑘 = 𝑀 ∨ 𝑘 = 𝑁 ∨ 𝑘 = 𝑂)) | |
22 | 21 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) → (𝑘 = 𝑀 ∨ 𝑘 = 𝑁 ∨ 𝑘 = 𝑂)) |
23 | 10, 15, 20, 22 | mpjao3dan 1432 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀, 𝑁, 𝑂}) → 𝐴 ∈ 𝐵) |
24 | gsumtp.7 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝑂) | |
25 | gsumtp.6 | . . . 4 ⊢ (𝜑 → 𝑁 ≠ 𝑂) | |
26 | disjprsn 4739 | . . . 4 ⊢ ((𝑀 ≠ 𝑂 ∧ 𝑁 ≠ 𝑂) → ({𝑀, 𝑁} ∩ {𝑂}) = ∅) | |
27 | 24, 25, 26 | syl2anc 583 | . . 3 ⊢ (𝜑 → ({𝑀, 𝑁} ∩ {𝑂}) = ∅) |
28 | df-tp 4653 | . . . 4 ⊢ {𝑀, 𝑁, 𝑂} = ({𝑀, 𝑁} ∪ {𝑂}) | |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑀, 𝑁, 𝑂} = ({𝑀, 𝑁} ∪ {𝑂})) |
30 | 1, 2, 3, 5, 23, 27, 29 | gsummptfidmsplit 19967 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁, 𝑂} ↦ 𝐴)) = ((𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) + (𝐺 Σg (𝑘 ∈ {𝑂} ↦ 𝐴)))) |
31 | gsumtp.2 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
32 | gsumtp.3 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊) | |
33 | gsumtp.5 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝑁) | |
34 | 1, 2, 6, 11 | gsumpr 19992 | . . . 4 ⊢ ((𝐺 ∈ CMnd ∧ (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊 ∧ 𝑀 ≠ 𝑁) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷)) |
35 | 3, 31, 32, 33, 8, 13, 34 | syl132anc 1388 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷)) |
36 | 3 | cmnmndd 19841 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
37 | gsumtp.4 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑋) | |
38 | 16 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑂) → 𝐴 = 𝐸) |
39 | 1, 36, 37, 18, 38 | gsumsnd 19989 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑂} ↦ 𝐴)) = 𝐸) |
40 | 35, 39 | oveq12d 7463 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) + (𝐺 Σg (𝑘 ∈ {𝑂} ↦ 𝐴))) = ((𝐶 + 𝐷) + 𝐸)) |
41 | 30, 40 | eqtrd 2774 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁, 𝑂} ↦ 𝐴)) = ((𝐶 + 𝐷) + 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∪ cun 3968 ∩ cin 3969 ∅c0 4347 {csn 4648 {cpr 4650 {ctp 4652 ↦ cmpt 5252 ‘cfv 6572 (class class class)co 7445 Fincfn 8999 Basecbs 17253 +gcplusg 17306 Σg cgsu 17495 CMndccmn 19817 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-seq 14049 df-hash 14376 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-0g 17496 df-gsum 17497 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 |
This theorem is referenced by: evl1deg2 33559 |
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