ipasslem3 - Metamath Proof Explorer

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Theorem ipasslem3 31194

Description: Lemma for ipassi 31202. Show the inner product associative law for all integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ip1i.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip1i.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
ip1i.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
ip1i.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
ip1i.9	⊢ 𝑈 ∈ CPreHil_OLD
ipasslem1.b	⊢ 𝐵 ∈ 𝑋

**Assertion**
Ref	Expression
ipasslem3	⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

**Proof of Theorem ipasslem3**
Step	Hyp	Ref	Expression
1		elznn0nn 12609	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
2		ip1i.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
3		ip1i.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4		ip1i.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
5		ip1i.7	. . . 4 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
6		ip1i.9	. . . 4 ⊢ 𝑈 ∈ CPreHil_OLD
7		ipasslem1.b	. . . 4 ⊢ 𝐵 ∈ 𝑋
8	2, 3, 4, 5, 6, 7	ipasslem1 31192	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
9		nnnn0 12515	. . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
10	2, 3, 4, 5, 6, 7	ipasslem2 31193	. . . . . 6 ⊢ ((-𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
11	9, 10	sylan 591	. . . . 5 ⊢ ((-𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
12	11	adantll 726	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
13		recn 11194	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
14	13	negnegd 11564	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → --𝑁 = 𝑁)
15	14	oveq1d 7425	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (--𝑁𝑆𝐴) = (𝑁𝑆𝐴))
16	15	oveq1d 7425	. . . . 5 ⊢ (𝑁 ∈ ℝ → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
17	16	ad2antrr 738	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
18	14	oveq1d 7425	. . . . 5 ⊢ (𝑁 ∈ ℝ → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
19	18	ad2antrr 738	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
20	12, 17, 19	3eqtr3d 2806	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
21	8, 20	jaoian 971	. 2 ⊢ (((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
22	1, 21	sylanb 592	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ‘cfv 6536 (class class class)co 7410 ℝcr 11103 · cmul 11109 -cneg 11446 ℕcn 12237 ℕ₀cn0 12508 ℤcz 12595 +_𝑣 cpv 30946 BaseSetcba 30947 ·_𝑠OLD cns 30948 ·_𝑖OLDcdip 31061 CPreHil_OLDccphlo 31173

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

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-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-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 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-n0 12509 df-z 12596 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-grpo 30854 df-gid 30855 df-ginv 30856 df-ablo 30906 df-vc 30920 df-nv 30953 df-va 30956 df-ba 30957 df-sm 30958 df-0v 30959 df-nmcv 30961 df-dip 31062 df-ph 31174

This theorem is used by: ipasslem5 31196

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