Theorem gsumpropd 13689

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 2235	. . . . . . 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 6086	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
7	1, 2, 3, 4, 6	grpidpropdg 13671	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
8	7	eqeq2d 2246	. . . . 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 10840	. . . . . . . . 9 ⊢ (𝜑 → seq𝑚((+g‘𝐺), 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
11	10	fveq1d 5677	. . . . . . . 8 ⊢ (𝜑 → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
12	11	eqeq2d 2246	. . . . . . 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 2545	. . . . 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 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))))
16	9, 15	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)))))
17	16	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)))))
18		eqid 2234	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
19		eqid 2234	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
20		eqid 2234	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
21		gsumpropd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
22		eqidd 2235	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
23	18, 19, 20, 3, 21, 22	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)))))
24		eqid 2234	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
25		eqid 2234	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
26		eqid 2234	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
27	24, 25, 26, 4, 21, 22	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)))))
28	17, 23, 27	3eqtr4d 2277	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523  ∅c0 3512  dom cdm 4754  ℩cio 5315  ‘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-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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-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-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-id 4419  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-recs 6549  df-frec 6635  df-neg 8463  df-inn 9255  df-z 9595  df-uz 9872  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555  df-igsum 13556

This theorem is referenced by: (None)