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Mirrors > Home > MPE Home > Th. List > gsummgmpropd | Structured version Visualization version GIF version |
Description: A stronger version of gsumpropd 18150 if at least one of the involved structures is a magma, see gsumpropd2 18152. (Contributed by AV, 31-Jan-2020.) |
Ref | Expression |
---|---|
gsummgmpropd.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
gsummgmpropd.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
gsummgmpropd.h | ⊢ (𝜑 → 𝐻 ∈ 𝑋) |
gsummgmpropd.b | ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) |
gsummgmpropd.m | ⊢ (𝜑 → 𝐺 ∈ Mgm) |
gsummgmpropd.e | ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) |
gsummgmpropd.n | ⊢ (𝜑 → Fun 𝐹) |
gsummgmpropd.r | ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) |
Ref | Expression |
---|---|
gsummgmpropd | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummgmpropd.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
2 | gsummgmpropd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
3 | gsummgmpropd.h | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
4 | gsummgmpropd.b | . 2 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) | |
5 | gsummgmpropd.m | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mgm) | |
6 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 6, 7 | mgmcl 18117 | . . . . 5 ⊢ ((𝐺 ∈ Mgm ∧ 𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) |
9 | 8 | 3expib 1124 | . . . 4 ⊢ (𝐺 ∈ Mgm → ((𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))) |
10 | 5, 9 | syl 17 | . . 3 ⊢ (𝜑 → ((𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))) |
11 | 10 | imp 410 | . 2 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) |
12 | gsummgmpropd.e | . 2 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) | |
13 | gsummgmpropd.n | . 2 ⊢ (𝜑 → Fun 𝐹) | |
14 | gsummgmpropd.r | . 2 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) | |
15 | 1, 2, 3, 4, 11, 12, 13, 14 | gsumpropd2 18152 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 ran crn 5552 Fun wfun 6374 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 Σg cgsu 16945 Mgmcmgm 18112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-seq 13575 df-0g 16946 df-gsum 16947 df-mgm 18114 |
This theorem is referenced by: gsumply1subr 21155 |
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