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Mirrors > Home > MPE Home > Th. List > gsummgmpropd | Structured version Visualization version GIF version |
Description: A stronger version of gsumpropd 18666 if at least one of the involved structures is a magma, see gsumpropd2 18668. (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 2725 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2725 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 6, 7 | mgmcl 18631 | . . . . 5 ⊢ ((𝐺 ∈ Mgm ∧ 𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) |
9 | 8 | 3expib 1119 | . . . 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 405 | . 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 18668 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3946 ran crn 5682 Fun wfun 6547 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 +gcplusg 17261 Σg cgsu 17450 Mgmcmgm 18626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-n0 12520 df-z 12606 df-uz 12870 df-fz 13534 df-seq 14017 df-0g 17451 df-gsum 17452 df-mgm 18628 |
This theorem is referenced by: gsumply1subr 22215 |
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