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Theorem gsumpropd2 12979
Description: A stronger version of gsumpropd 12978, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 12980. (Contributed by Thierry Arnoux, 28-Jun-2017.)
Hypotheses
Ref Expression
gsumpropd2.f (𝜑𝐹𝑉)
gsumpropd2.g (𝜑𝐺𝑊)
gsumpropd2.h (𝜑𝐻𝑋)
gsumpropd2.b (𝜑 → (Base‘𝐺) = (Base‘𝐻))
gsumpropd2.c ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
gsumpropd2.e ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
gsumpropd2.n (𝜑 → Fun 𝐹)
gsumpropd2.r (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
Assertion
Ref Expression
gsumpropd2 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Distinct variable groups:   𝐹,𝑠,𝑡   𝐺,𝑠,𝑡   𝐻,𝑠,𝑡   𝜑,𝑠,𝑡
Allowed substitution hints:   𝑉(𝑡,𝑠)   𝑊(𝑡,𝑠)   𝑋(𝑡,𝑠)

Proof of Theorem gsumpropd2
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2194 . . . . . . 7 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
2 gsumpropd2.b . . . . . . 7 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
3 gsumpropd2.g . . . . . . 7 (𝜑𝐺𝑊)
4 gsumpropd2.h . . . . . . 7 (𝜑𝐻𝑋)
5 gsumpropd2.e . . . . . . 7 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
61, 2, 3, 4, 5grpidpropdg 12960 . . . . . 6 (𝜑 → (0g𝐺) = (0g𝐻))
76eqeq2d 2205 . . . . 5 (𝜑 → (𝑥 = (0g𝐺) ↔ 𝑥 = (0g𝐻)))
87anbi2d 464 . . . 4 (𝜑 → ((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐺)) ↔ (dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐻))))
9 simprl 529 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ𝑚))
10 gsumpropd2.r . . . . . . . . . . . 12 (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
1110ad2antrr 488 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
12 gsumpropd2.n . . . . . . . . . . . . 13 (𝜑 → Fun 𝐹)
1312ad2antrr 488 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
14 simpr 110 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
15 simplrr 536 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
1614, 15eleqtrrd 2273 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
17 fvelrn 5690 . . . . . . . . . . . 12 ((Fun 𝐹𝑠 ∈ dom 𝐹) → (𝐹𝑠) ∈ ran 𝐹)
1813, 16, 17syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ ran 𝐹)
1911, 18sseldd 3181 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹𝑠) ∈ (Base‘𝐺))
20 gsumpropd2.f . . . . . . . . . . 11 (𝜑𝐹𝑉)
2120adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝐹𝑉)
22 plusgslid 12733 . . . . . . . . . . . . 13 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
2322slotex 12648 . . . . . . . . . . . 12 (𝐺𝑊 → (+g𝐺) ∈ V)
243, 23syl 14 . . . . . . . . . . 11 (𝜑 → (+g𝐺) ∈ V)
2524adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (+g𝐺) ∈ V)
2622slotex 12648 . . . . . . . . . . . 12 (𝐻𝑋 → (+g𝐻) ∈ V)
274, 26syl 14 . . . . . . . . . . 11 (𝜑 → (+g𝐻) ∈ V)
2827adantr 276 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (+g𝐻) ∈ V)
29 gsumpropd2.c . . . . . . . . . . 11 ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
3029adantlr 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) ∈ (Base‘𝐺))
315adantlr 477 . . . . . . . . . 10 (((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g𝐺)𝑡) = (𝑠(+g𝐻)𝑡))
329, 19, 21, 25, 28, 30, 31seqfeq4g 10605 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g𝐺), 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
3332eqeq2d 2205 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (ℤ𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3433anassrs 400 . . . . . . 7 (((𝜑𝑛 ∈ (ℤ𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
3534pm5.32da 452 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3635rexbidva 2491 . . . . 5 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
3736exbidv 1836 . . . 4 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
388, 37orbi12d 794 . . 3 (𝜑 → (((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛))) ↔ ((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐻)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))))
3938iotabidv 5238 . 2 (𝜑 → (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)))) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐻)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))))
40 eqid 2193 . . 3 (Base‘𝐺) = (Base‘𝐺)
41 eqid 2193 . . 3 (0g𝐺) = (0g𝐺)
42 eqid 2193 . . 3 (+g𝐺) = (+g𝐺)
43 eqidd 2194 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
4440, 41, 42, 3, 20, 43igsumvalx 12975 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐺), 𝐹)‘𝑛)))))
45 eqid 2193 . . 3 (Base‘𝐻) = (Base‘𝐻)
46 eqid 2193 . . 3 (0g𝐻) = (0g𝐻)
47 eqid 2193 . . 3 (+g𝐻) = (+g𝐻)
4845, 46, 47, 4, 20, 43igsumvalx 12975 . 2 (𝜑 → (𝐻 Σg 𝐹) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g𝐻)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))))
4939, 44, 483eqtr4d 2236 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wex 1503  wcel 2164  wrex 2473  Vcvv 2760  wss 3154  c0 3447  dom cdm 4660  ran crn 4661  cio 5214  Fun wfun 5249  cfv 5255  (class class class)co 5919  cuz 9595  ...cfz 10077  seqcseq 10521  Basecbs 12621  +gcplusg 12698  0gc0g 12870   Σg cgsu 12871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-igsum 12873
This theorem is referenced by:  gsummgmpropd  12980
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