ipasslem3 - Metamath Proof Explorer

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Theorem ipasslem3 30857

Description: Lemma for ipassi 30865. 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 12500	. 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 30855	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
9		nnnn0 12406	. . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
10	2, 3, 4, 5, 6, 7	ipasslem2 30856	. . . . . 6 ⊢ ((-𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
11	9, 10	sylan 580	. . . . 5 ⊢ ((-𝑁 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
12	11	adantll 714	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
13		recn 11114	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
14	13	negnegd 11481	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → --𝑁 = 𝑁)
15	14	oveq1d 7371	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (--𝑁𝑆𝐴) = (𝑁𝑆𝐴))
16	15	oveq1d 7371	. . . . 5 ⊢ (𝑁 ∈ ℝ → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
17	16	ad2antrr 726	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = ((𝑁𝑆𝐴)𝑃𝐵))
18	14	oveq1d 7371	. . . . 5 ⊢ (𝑁 ∈ ℝ → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
19	18	ad2antrr 726	. . . 4 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → (--𝑁 · (𝐴𝑃𝐵)) = (𝑁 · (𝐴𝑃𝐵)))
20	12, 17, 19	3eqtr3d 2777	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
21	8, 20	jaoian 958	. 2 ⊢ (((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
22	1, 21	sylanb 581	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ‘cfv 6490 (class class class)co 7356 ℝcr 11023 · cmul 11029 -cneg 11363 ℕcn 12143 ℕ₀cn0 12399 ℤcz 12486 +_𝑣 cpv 30609 BaseSetcba 30610 ·_𝑠OLD cns 30611 ·_𝑖OLDcdip 30724 CPreHil_OLDccphlo 30836

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-sum 15608 df-grpo 30517 df-gid 30518 df-ginv 30519 df-ablo 30569 df-vc 30583 df-nv 30616 df-va 30619 df-ba 30620 df-sm 30621 df-0v 30622 df-nmcv 30624 df-dip 30725 df-ph 30837

This theorem is referenced by: ipasslem5 30859

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