crosspdotsumi - Mathbox for Jiamin Zhao

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Theorem crosspdotsumi 50673

Description: Expand the group sum over (1...3) into an explicit three-term sum. (A helper for crosspdoti 50674.) (Contributed by Jiamin Zhao, 1-Aug-2026.)

**Hypotheses**
Ref	Expression
crosspdotsumi.1	⊢ 𝐴 ∈ (ℝ ↑_m (1...3))
crosspdotsumi.2	⊢ 𝐵 ∈ (ℝ ↑_m (1...3))
crosspdotsumi.3	⊢ 𝐶 ∈ (ℝ ↑_m (1...3))

**Assertion**
Ref	Expression
crosspdotsumi	⊢ (ℝ_fld Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝐶,𝑘

**Proof of Theorem crosspdotsumi**
Step	Hyp	Ref	Expression
1		df-refld 21764	. . 3 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
2	1	oveq1i 7420	. 2 ⊢ (ℝ_fld Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = ((ℂ_fld ↾_s ℝ) Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))))
3		crosspdotsumi.1	. . 3 ⊢ 𝐴 ∈ (ℝ ↑_m (1...3))
4		fzfid 14014	. . . 4 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → (1...3) ∈ Fin)
5		elmapi 8842	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → 𝐴:(1...3)⟶ℝ)
6	5	ffvelcdmda 7079	. . . . 5 ⊢ ((𝐴 ∈ (ℝ ↑_m (1...3)) ∧ 𝑘 ∈ (1...3)) → (𝐴‘𝑘) ∈ ℝ)
7		crosspdotsumi.2	. . . . . . . . 9 ⊢ 𝐵 ∈ (ℝ ↑_m (1...3))
8		crosspdotsumi.3	. . . . . . . . 9 ⊢ 𝐶 ∈ (ℝ ↑_m (1...3))
9	7, 8	crosspcli 50668	. . . . . . . 8 ⊢ (𝐵⊠𝐶) ∈ (ℝ ↑_m (1...3))
10		elmapi 8842	. . . . . . . 8 ⊢ ((𝐵⊠𝐶) ∈ (ℝ ↑_m (1...3)) → (𝐵⊠𝐶):(1...3)⟶ℝ)
11	9, 10	ax-mp 5	. . . . . . 7 ⊢ (𝐵⊠𝐶):(1...3)⟶ℝ
12	11	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ↑_m (1...3)) ∧ 𝑘 ∈ (1...3)) → (𝐵⊠𝐶):(1...3)⟶ℝ)
13		simpr 489	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ↑_m (1...3)) ∧ 𝑘 ∈ (1...3)) → 𝑘 ∈ (1...3))
14	12, 13	ffvelcdmd 7080	. . . . 5 ⊢ ((𝐴 ∈ (ℝ ↑_m (1...3)) ∧ 𝑘 ∈ (1...3)) → ((𝐵⊠𝐶)‘𝑘) ∈ ℝ)
15	6, 14	remulcld 11243	. . . 4 ⊢ ((𝐴 ∈ (ℝ ↑_m (1...3)) ∧ 𝑘 ∈ (1...3)) → ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) ∈ ℝ)
16	4, 15	regsumfsum 21594	. . 3 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → ((ℂ_fld ↾_s ℝ) Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = Σ𝑘 ∈ (1...3)((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))
17	3, 16	ax-mp 5	. 2 ⊢ ((ℂ_fld ↾_s ℝ) Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = Σ𝑘 ∈ (1...3)((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))
18		1p2e3 12387	. . . . . . 7 ⊢ (1 + 2) = 3
19	18	eqcomi 2772	. . . . . 6 ⊢ 3 = (1 + 2)
20	19	oveq2i 7421	. . . . 5 ⊢ (1...3) = (1...(1 + 2))
21		1z 12628	. . . . . 6 ⊢ 1 ∈ ℤ
22		fztp 13613	. . . . . 6 ⊢ (1 ∈ ℤ → (1...(1 + 2)) = {1, (1 + 1), (1 + 2)})
23	21, 22	ax-mp 5	. . . . 5 ⊢ (1...(1 + 2)) = {1, (1 + 1), (1 + 2)}
24	20, 23	eqtri 2786	. . . 4 ⊢ (1...3) = {1, (1 + 1), (1 + 2)}
25	24	sumeq1i 15753	. . 3 ⊢ Σ𝑘 ∈ (1...3)((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = Σ𝑘 ∈ {1, (1 + 1), (1 + 2)} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))
26		eqidd 2764	. . . . . 6 ⊢ (1 ∈ ℤ → 1 = 1)
27		1p1e2 12368	. . . . . . 7 ⊢ (1 + 1) = 2
28	27	a1i 11	. . . . . 6 ⊢ (1 ∈ ℤ → (1 + 1) = 2)
29	18	a1i 11	. . . . . 6 ⊢ (1 ∈ ℤ → (1 + 2) = 3)
30	26, 28, 29	tpeq123d 4714	. . . . 5 ⊢ (1 ∈ ℤ → {1, (1 + 1), (1 + 2)} = {1, 2, 3})
31	21, 30	ax-mp 5	. . . 4 ⊢ {1, (1 + 1), (1 + 2)} = {1, 2, 3}
32	31	sumeq1i 15753	. . 3 ⊢ Σ𝑘 ∈ {1, (1 + 1), (1 + 2)} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = Σ𝑘 ∈ {1, 2, 3} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘))
33		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 1 → (𝐴‘𝑘) = (𝐴‘1))
34		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 1 → ((𝐵⊠𝐶)‘𝑘) = ((𝐵⊠𝐶)‘1))
35	33, 34	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 1 → ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = ((𝐴‘1) · ((𝐵⊠𝐶)‘1)))
36		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 2 → (𝐴‘𝑘) = (𝐴‘2))
37		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 2 → ((𝐵⊠𝐶)‘𝑘) = ((𝐵⊠𝐶)‘2))
38	36, 37	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 2 → ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = ((𝐴‘2) · ((𝐵⊠𝐶)‘2)))
39		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 3 → (𝐴‘𝑘) = (𝐴‘3))
40		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 3 → ((𝐵⊠𝐶)‘𝑘) = ((𝐵⊠𝐶)‘3))
41	39, 40	oveq12d 7428	. . . . . 6 ⊢ (𝑘 = 3 → ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = ((𝐴‘3) · ((𝐵⊠𝐶)‘3)))
42	3	rr3fv1cli 50656	. . . . . . . . . 10 ⊢ (𝐴‘1) ∈ ℝ
43	9	rr3fv1cli 50656	. . . . . . . . . 10 ⊢ ((𝐵⊠𝐶)‘1) ∈ ℝ
44	42, 43	remulcli 11229	. . . . . . . . 9 ⊢ ((𝐴‘1) · ((𝐵⊠𝐶)‘1)) ∈ ℝ
45	44	recni 11227	. . . . . . . 8 ⊢ ((𝐴‘1) · ((𝐵⊠𝐶)‘1)) ∈ ℂ
46	45	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → ((𝐴‘1) · ((𝐵⊠𝐶)‘1)) ∈ ℂ)
47	3	rr3fv2cli 50657	. . . . . . . . . 10 ⊢ (𝐴‘2) ∈ ℝ
48	9	rr3fv2cli 50657	. . . . . . . . . 10 ⊢ ((𝐵⊠𝐶)‘2) ∈ ℝ
49	47, 48	remulcli 11229	. . . . . . . . 9 ⊢ ((𝐴‘2) · ((𝐵⊠𝐶)‘2)) ∈ ℝ
50	49	recni 11227	. . . . . . . 8 ⊢ ((𝐴‘2) · ((𝐵⊠𝐶)‘2)) ∈ ℂ
51	50	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → ((𝐴‘2) · ((𝐵⊠𝐶)‘2)) ∈ ℂ)
52	3	rr3fv3cli 50658	. . . . . . . . . 10 ⊢ (𝐴‘3) ∈ ℝ
53	9	rr3fv3cli 50658	. . . . . . . . . 10 ⊢ ((𝐵⊠𝐶)‘3) ∈ ℝ
54	52, 53	remulcli 11229	. . . . . . . . 9 ⊢ ((𝐴‘3) · ((𝐵⊠𝐶)‘3)) ∈ ℝ
55	54	recni 11227	. . . . . . . 8 ⊢ ((𝐴‘3) · ((𝐵⊠𝐶)‘3)) ∈ ℂ
56	55	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → ((𝐴‘3) · ((𝐵⊠𝐶)‘3)) ∈ ℂ)
57	46, 51, 56	3jca 1146	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) ∈ ℂ ∧ ((𝐴‘2) · ((𝐵⊠𝐶)‘2)) ∈ ℂ ∧ ((𝐴‘3) · ((𝐵⊠𝐶)‘3)) ∈ ℂ))
58		2z 12630	. . . . . . . 8 ⊢ 2 ∈ ℤ
59		3z 12631	. . . . . . . 8 ⊢ 3 ∈ ℤ
60	21, 58, 59	3pm3.2i 1358	. . . . . . 7 ⊢ (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ)
61	60	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → (1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 3 ∈ ℤ))
62		1ne2 12455	. . . . . . 7 ⊢ 1 ≠ 2
63	62	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → 1 ≠ 2)
64		1ne3 50650	. . . . . . 7 ⊢ 1 ≠ 3
65	64	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → 1 ≠ 3)
66		2ne3 50651	. . . . . . 7 ⊢ 2 ≠ 3
67	66	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → 2 ≠ 3)
68	35, 38, 41, 57, 61, 63, 65, 67	sumtp 15805	. . . . 5 ⊢ (𝐴 ∈ (ℝ ↑_m (1...3)) → Σ𝑘 ∈ {1, 2, 3} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = ((((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + ((𝐴‘2) · ((𝐵⊠𝐶)‘2))) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))
69	3, 68	ax-mp 5	. . . 4 ⊢ Σ𝑘 ∈ {1, 2, 3} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = ((((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + ((𝐴‘2) · ((𝐵⊠𝐶)‘2))) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3)))
70	45, 50, 55	addassi 11223	. . . 4 ⊢ ((((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + ((𝐴‘2) · ((𝐵⊠𝐶)‘2))) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))
71	69, 70	eqtri 2786	. . 3 ⊢ Σ𝑘 ∈ {1, 2, 3} ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))
72	25, 32, 71	3eqtri 2790	. 2 ⊢ Σ𝑘 ∈ (1...3)((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))
73	2, 17, 72	3eqtri 2790	1 ⊢ (ℝ_fld Σ_g (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 {ctp 4593 ↦ cmpt 5192 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8820 ℂcc 11102 ℝcr 11103 1c1 11105 + caddc 11107 · cmul 11109 2c2 12299 3c3 12300 ℤcz 12595 ...cfz 13539 Σcsu 15742 ↾_s cress 17294 Σ_g cgsu 17497 ℂ_fldccnfld 21531 ℝ_fldcrefld 21763 ⊠ccrossp 50659

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-rp 13021 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-clim 15544 df-sum 15743 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-0g 17498 df-gsum 17499 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-grp 19007 df-minusg 19008 df-cntz 19391 df-cmn 19856 df-abl 19857 df-mgp 20221 df-ur 20268 df-ring 20321 df-cring 20322 df-cnfld 21532 df-refld 21764 df-crossp 50660

This theorem is used by: crosspdoti 50674

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