ipasslem3 - Metamath Proof Explorer

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Theorem ipasslem3 31346

Description: Lemma for ipassi 31354. 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 12654	. 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 31344	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
9		nnnn0 12560	. . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
10	2, 3, 4, 5, 6, 7	ipasslem2 31345	. . . . . 6 ⊢ ((-𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
11	9, 10	sylan 592	. . . . 5 ⊢ ((-𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
12	11	adantll 727	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
13		recn 11239	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
14	13	negnegd 11609	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → --𝑁 = 𝑁)
15	14	oveq1d 7431	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (--𝑁𝑆𝐴) = (𝑁𝑆𝐴))
16	15	oveq1d 7431	. . . . 5 ⊢ (𝑁 ∈ ℝ → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
17	16	ad2antrr 739	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
18	14	oveq1d 7431	. . . . 5 ⊢ (𝑁 ∈ ℝ → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
19	18	ad2antrr 739	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
20	12, 17, 19	3eqtr3d 2803	. . 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 593	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ‘cfv 6535 (class class class)co 7416 ℝcr 11148 · cmul 11154 -cneg 11491 ℕcn 12282 ℕ₀cn0 12553 ℤcz 12640 +_𝑣 cpv 31098 BaseSetcba 31099 ·_𝑠OLD cns 31100 ·_𝑖OLDcdip 31213 CPreHil_OLDccphlo 31325

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9419 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-n0 12554 df-z 12641 df-uz 12913 df-rp 13068 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-clim 15600 df-sum 15799 df-grpo 31006 df-gid 31007 df-ginv 31008 df-ablo 31058 df-vc 31072 df-nv 31105 df-va 31108 df-ba 31109 df-sm 31110 df-0v 31111 df-nmcv 31113 df-dip 31214 df-ph 31326

This theorem is used by: ipasslem5 31348

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