dpjeq - Metamath Proof Explorer

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Theorem dpjeq 19973

Description: Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)

**Hypotheses**
Ref	Expression
dpjfval.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dpjfval.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p	⊢ 𝑃 = (𝐺dProj𝑆)
dpjidcl.3	⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆))
dpjidcl.0	⊢ 0 = (0_g‘𝐺)
dpjidcl.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
dpjeq.c	⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊)

**Assertion**
Ref	Expression
dpjeq	⊢ (𝜑 → (𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))

Distinct variable groups: 𝑥,ℎ, 0 ℎ,𝑖,𝐺,𝑥 𝑃,ℎ,𝑥 𝜑,𝑖,𝑥 𝐶,ℎ ℎ,𝐼,𝑖,𝑥 𝑥,𝑊 𝐴,ℎ,𝑥 𝑆,ℎ,𝑖,𝑥 Allowed substitution hints: 𝜑(ℎ) 𝐴(𝑖) 𝐶(𝑥,𝑖) 𝑃(𝑖) 𝑊(ℎ,𝑖) 0 (𝑖)

**Proof of Theorem dpjeq**
Step	Hyp	Ref	Expression
1		dpjfval.1	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dpjfval.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
3		dpjfval.p	. . . . 5 ⊢ 𝑃 = (𝐺dProj𝑆)
4		dpjidcl.3	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆))
5		dpjidcl.0	. . . . 5 ⊢ 0 = (0_g‘𝐺)
6		dpjidcl.w	. . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
7	1, 2, 3, 4, 5, 6	dpjidcl 19972	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))))
8	7	simprd 495	. . 3 ⊢ (𝜑 → 𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))
9	8	eqeq1d 2733	. 2 ⊢ (𝜑 → (𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶))))
10	7	simpld 494	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊)
11		dpjeq.c	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐶) ∈ 𝑊)
12	5, 6, 1, 2, 10, 11	dprdf11 19937	. 2 ⊢ (𝜑 → ((𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶)))
13		fvex 6835	. . . 4 ⊢ ((𝑃‘𝑥)‘𝐴) ∈ V
14	13	rgenw 3051	. . 3 ⊢ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) ∈ V
15		mpteqb 6948	. . 3 ⊢ (∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) ∈ V → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))
16	14, 15	mp1i 13	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))
17	9, 12, 16	3bitrd 305	1 ⊢ (𝜑 → (𝐴 = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 𝐶)) ↔ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) = 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1541 ∈ wcel 2111 ∀wral 3047 {crab 3395 Vcvv 3436 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 Xcixp 8821 finSupp cfsupp 9245 0_gc0g 17343 Σ_g cgsu 17344 DProd cdprd 19907 dProjcdpj 19908

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-0g 17345 df-gsum 17346 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-gim 19171 df-cntz 19229 df-oppg 19258 df-lsm 19548 df-pj1 19549 df-cmn 19694 df-dprd 19909 df-dpj 19910

This theorem is referenced by: dpjrid 19976 dchrptlem3 27204

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