dpjeq - Metamath Proof Explorer

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Theorem dpjeq 19977

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 19976	. . . 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 2726	. 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 19941	. 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 6895	. . . 4 ⊢ ((𝑃‘𝑥)‘𝐴) ∈ V
14	13	rgenw 3057	. . 3 ⊢ ∀𝑥 ∈ 𝐼 ((𝑃‘𝑥)‘𝐴) ∈ V
15		mpteqb 7008	. . 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 205 = wceq 1533 ∈ wcel 2098 ∀wral 3053 {crab 3424 Vcvv 3466 class class class wbr 5139 ↦ cmpt 5222 dom cdm 5667 ‘cfv 6534 (class class class)co 7402 Xcixp 8888 finSupp cfsupp 9358 0_gc0g 17390 Σ_g cgsu 17391 DProd cdprd 19911 dProjcdpj 19912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-gsum 17393 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-gim 19180 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-pj1 19553 df-cmn 19698 df-dprd 19913 df-dpj 19914

This theorem is referenced by: dpjrid 19980 dchrptlem3 27140

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