ipasslem3 - Metamath Proof Explorer

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Theorem ipasslem3 28771

Description: Lemma for ipassi 28779. 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 12079	. 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 28769	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
9		nnnn0 11986	. . . . . 6 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℕ₀)
10	2, 3, 4, 5, 6, 7	ipasslem2 28770	. . . . . 6 ⊢ ((-𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋) → ((--𝑁𝑆𝐴)𝑃𝐵) = (--𝑁 · (𝐴𝑃𝐵)))
11	9, 10	sylan 583	. . . . 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 10708	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
14	13	negnegd 11069	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → --𝑁 = 𝑁)
15	14	oveq1d 7188	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (--𝑁𝑆𝐴) = (𝑁𝑆𝐴))
16	15	oveq1d 7188	. . . . 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 7188	. . . . 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 2782	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
21	8, 20	jaoian 956	. 2 ⊢ (((𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))
22	1, 21	sylanb 584	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑁𝑆𝐴)𝑃𝐵) = (𝑁 · (𝐴𝑃𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 ℝcr 10617 · cmul 10623 -cneg 10952 ℕcn 11719 ℕ₀cn0 11979 ℤcz 12065 +_𝑣 cpv 28523 BaseSetcba 28524 ·_𝑠OLD cns 28525 ·_𝑖OLDcdip 28638 CPreHil_OLDccphlo 28750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-oi 9050 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-n0 11980 df-z 12066 df-uz 12328 df-rp 12476 df-fz 12985 df-fzo 13128 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-clim 14938 df-sum 15139 df-grpo 28431 df-gid 28432 df-ginv 28433 df-ablo 28483 df-vc 28497 df-nv 28530 df-va 28533 df-ba 28534 df-sm 28535 df-0v 28536 df-nmcv 28538 df-dip 28639 df-ph 28751

This theorem is referenced by: ipasslem5 28773

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