Theorem gsumpropd2 13690

Description: A stronger version of gsumpropd 13689, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 13691. (Contributed by Thierry Arnoux, 28-Jun-2017.)

Hypotheses

Ref	Expression
gsumpropd2.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
gsumpropd2.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
gsumpropd2.h	⊢ (𝜑 → 𝐻 ∈ 𝑋)
gsumpropd2.b	⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
gsumpropd2.c	⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
gsumpropd2.e	⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)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.

Step	Hyp	Ref	Expression
1		eqidd 2235	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
2		gsumpropd2.b	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
3		gsumpropd2.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		gsumpropd2.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
5		gsumpropd2.e	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))
6	1, 2, 3, 4, 5	grpidpropdg 13671	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
7	6	eqeq2d 2246	. . . . 5 ⊢ (𝜑 → (𝑥 = (0g‘𝐺) ↔ 𝑥 = (0g‘𝐻)))
8	7	anbi2d 464	. . . 4 ⊢ (𝜑 → ((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐺)) ↔ (dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐻))))
9		simprl 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ≥‘𝑚))
10		gsumpropd2.r	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
11	10	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
12		gsumpropd2.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → Fun 𝐹)
13	12	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
14		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
15		simplrr 538	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
16	14, 15	eleqtrrd 2314	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
17		fvelrn 5813	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ dom 𝐹) → (𝐹‘𝑠) ∈ ran 𝐹)
18	13, 16, 17	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ ran 𝐹)
19	11, 18	sseldd 3243	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ (Base‘𝐺))
20		gsumpropd2.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
21	20	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝐹 ∈ 𝑉)
22		plusgslid 13409	. . . . . . . . . . . . 13 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
23	22	slotex 13323	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝑊 → (+g‘𝐺) ∈ V)
24	3, 23	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (+g‘𝐺) ∈ V)
25	24	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (+g‘𝐺) ∈ V)
26	22	slotex 13323	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ 𝑋 → (+g‘𝐻) ∈ V)
27	4, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (+g‘𝐻) ∈ V)
28	27	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (+g‘𝐻) ∈ V)
29		gsumpropd2.c	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
30	29	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
31	5	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))
32	9, 19, 21, 25, 28, 30, 31	seqfeq4g 10917	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
33	32	eqeq2d 2246	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
34	33	anassrs 400	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
35	34	pm5.32da 452	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
36	35	rexbidva 2541	. . . . 5 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
37	36	exbidv 1874	. . . 4 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
38	8, 37	orbi12d 801	. . 3 ⊢ (𝜑 → (((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛))) ↔ ((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
39	38	iotabidv 5340	. 2 ⊢ (𝜑 → (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)))) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
40		eqid 2234	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
41		eqid 2234	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
42		eqid 2234	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
43		eqidd 2235	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
44	40, 41, 42, 3, 20, 43	igsumvalx 13686	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)))))
45		eqid 2234	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
46		eqid 2234	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
47		eqid 2234	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
48	45, 46, 47, 4, 20, 43	igsumvalx 13686	. 2 ⊢ (𝜑 → (𝐻 Σg 𝐹) = (℩𝑥((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
49	39, 44, 48	3eqtr4d 2277	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523  Vcvv 2815   ⊆ wss 3214  ∅c0 3512  dom cdm 4754  ran crn 4755  ℩cio 5315  Fun wfun 5351  ‘cfv 5357  (class class class)co 6058  ℤ_≥cuz 9871  ...cfz 10361  seqcseq 10833  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553   Σ_g cgsu 13554

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-igsum 13556

This theorem is referenced by:  gsummgmpropd  13691