Theorem gsumpropd 13411

Description: The group sum depends only on the base set and additive operation. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)

Hypotheses

Ref	Expression
gsumpropd.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
gsumpropd.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
gsumpropd.h	⊢ (𝜑 → 𝐻 ∈ 𝑋)
gsumpropd.b	⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
gsumpropd.p	⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))

Assertion

Ref	Expression
gsumpropd	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Proof of Theorem gsumpropd

Dummy variables 𝑎 𝑏 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2230	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
2		gsumpropd.b	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
3		gsumpropd.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
4		gsumpropd.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
5		gsumpropd.p	. . . . . . . 8 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))
6	5	oveqdr 6022	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
7	1, 2, 3, 4, 6	grpidpropdg 13393	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
8	7	eqeq2d 2241	. . . . 5 ⊢ (𝜑 → (𝑥 = (0g‘𝐺) ↔ 𝑥 = (0g‘𝐻)))
9	8	anbi2d 464	. . . 4 ⊢ (𝜑 → ((dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐺)) ↔ (dom 𝐹 = ∅ ∧ 𝑥 = (0g‘𝐻))))
10	5	seqeq2d 10663	. . . . . . . . 9 ⊢ (𝜑 → seq𝑚((+g‘𝐺), 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
11	10	fveq1d 5625	. . . . . . . 8 ⊢ (𝜑 → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
12	11	eqeq2d 2241	. . . . . . 7 ⊢ (𝜑 → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
13	12	anbi2d 464	. . . . . 6 ⊢ (𝜑 → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
14	13	rexbidv 2531	. . . . 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))))
15	14	exbidv 1871	. . . 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))))
16	9, 15	orbi12d 798	. . 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)))))
17	16	iotabidv 5297	. 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)))))
18		eqid 2229	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		eqid 2229	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20		eqid 2229	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
21		gsumpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
22		eqidd 2230	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
23	18, 19, 20, 3, 21, 22	igsumvalx 13408	. 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)))))
24		eqid 2229	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
25		eqid 2229	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
26		eqid 2229	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
27	24, 25, 26, 4, 21, 22	igsumvalx 13408	. 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)))))
28	17, 23, 27	3eqtr4d 2272	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  ∅c0 3491  dom cdm 4716  ℩cio 5272  ‘cfv 5314  (class class class)co 5994  ℤ_≥cuz 9710  ...cfz 10192  seqcseq 10656  Basecbs 13018  +_gcplusg 13096  0_gc0g 13275   Σ_g cgsu 13276

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-recs 6441  df-frec 6527  df-neg 8308  df-inn 9099  df-z 9435  df-uz 9711  df-seqfrec 10657  df-ndx 13021  df-slot 13022  df-base 13024  df-0g 13277  df-igsum 13278

This theorem is referenced by: (None)