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Mirrors > Home > MPE Home > Th. List > gsummgmpropd | Structured version Visualization version GIF version |
Description: A stronger version of gsumpropd 17882 if at least one of the involved structures is a magma, see gsumpropd2 17884. (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 2821 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
7 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
8 | 6, 7 | mgmcl 17849 | . . . . 5 ⊢ ((𝐺 ∈ Mgm ∧ 𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺)) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) |
9 | 8 | 3expib 1118 | . . . 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 409 | . 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 17884 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ran crn 5550 Fun wfun 6343 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 Σg cgsu 16708 Mgmcmgm 17844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-0g 16709 df-gsum 16710 df-mgm 17846 |
This theorem is referenced by: gsumply1subr 20396 |
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