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Mirrors > Home > MPE Home > Th. List > gsummptfsadd | Structured version Visualization version GIF version |
Description: The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 9-Jul-2019.) |
Ref | Expression |
---|---|
gsummptfsadd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptfsadd.z | ⊢ 0 = (0g‘𝐺) |
gsummptfsadd.p | ⊢ + = (+g‘𝐺) |
gsummptfsadd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptfsadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsummptfsadd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
gsummptfsadd.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) |
gsummptfsadd.f | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
gsummptfsadd.h | ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) |
gsummptfsadd.w | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
gsummptfsadd.v | ⊢ (𝜑 → 𝐻 finSupp 0 ) |
Ref | Expression |
---|---|
gsummptfsadd | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptfsadd.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | gsummptfsadd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) | |
3 | gsummptfsadd.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐵) | |
4 | gsummptfsadd.f | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
5 | gsummptfsadd.h | . . . . 5 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ 𝐷)) | |
6 | 1, 2, 3, 4, 5 | offval2 7429 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) |
7 | 6 | eqcomd 2830 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷)) = (𝐹 ∘f + 𝐻)) |
8 | 7 | oveq2d 7175 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = (𝐺 Σg (𝐹 ∘f + 𝐻))) |
9 | gsummptfsadd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
10 | gsummptfsadd.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
11 | gsummptfsadd.p | . . 3 ⊢ + = (+g‘𝐺) | |
12 | gsummptfsadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
13 | 4, 2 | fmpt3d 6883 | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
14 | 5, 3 | fmpt3d 6883 | . . 3 ⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
15 | gsummptfsadd.w | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
16 | gsummptfsadd.v | . . 3 ⊢ (𝜑 → 𝐻 finSupp 0 ) | |
17 | 9, 10, 11, 12, 1, 13, 14, 15, 16 | gsumadd 19046 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
18 | 8, 17 | eqtrd 2859 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐶 + 𝐷))) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 finSupp cfsupp 8836 Basecbs 16486 +gcplusg 16568 0gc0g 16716 Σg cgsu 16717 CMndccmn 18909 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-gsum 16719 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-cntz 18450 df-cmn 18911 |
This theorem is referenced by: gsummptfidmadd 19048 frlmphl 20928 pm2mpghm 21427 lincsum 44491 |
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