ipassr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ipassr		Structured version Visualization version GIF version

Theorem ipassr 21979

Description: "Associative" law for second argument of inner product (compare ipass 21978). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)

**Hypotheses**
Ref	Expression
phlsrng.f	⊢ 𝐹 = (Scalar‘𝑊)
phllmhm.h	⊢ , = (·_𝑖‘𝑊)
phllmhm.v	⊢ 𝑉 = (Base‘𝑊)
ipdir.f	⊢ 𝐾 = (Base‘𝐹)
ipass.s	⊢ · = ( ·_𝑠 ‘𝑊)
ipass.p	⊢ × = (._r‘𝐹)
ipassr.i	⊢ ∗ = (*_𝑟‘𝐹)

**Assertion**
Ref	Expression
ipassr	⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))

**Proof of Theorem ipassr**
Step	Hyp	Ref	Expression
1		simpl 488	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ PreHil)
2		simpr3 1215	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐶 ∈ 𝐾)
3		simpr2 1214	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐵 ∈ 𝑉)
4		simpr1 1213	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐴 ∈ 𝑉)
5		phlsrng.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
6		phllmhm.h	. . . . . 6 ⊢ , = (·_𝑖‘𝑊)
7		phllmhm.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
8		ipdir.f	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
9		ipass.s	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑊)
10		ipass.p	. . . . . 6 ⊢ × = (._r‘𝐹)
11	5, 6, 7, 8, 9, 10	ipass 21978	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
12	1, 2, 3, 4, 11	syl13anc 1400	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ((𝐶 · 𝐵) , 𝐴) = (𝐶 × (𝐵 , 𝐴)))
13	12	fveq2d 6909	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = ( ∗ ‘(𝐶 × (𝐵 , 𝐴))))
14		phllmod 21963	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
15	14	adantr 486	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝑊 ∈ LMod)
16	7, 5, 9, 8	lmodvscl 21180	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐶 · 𝐵) ∈ 𝑉)
17	15, 2, 3, 16	syl3anc 1399	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐶 · 𝐵) ∈ 𝑉)
18		ipassr.i	. . . . 5 ⊢ ∗ = (*_𝑟‘𝐹)
19	5, 6, 7, 18	ipcj 21967	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐶 · 𝐵) ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
20	1, 17, 4, 19	syl3anc 1399	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘((𝐶 · 𝐵) , 𝐴)) = (𝐴 , (𝐶 · 𝐵)))
21	5	phlsrng 21964	. . . . 5 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)
22	21	adantr 486	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → 𝐹 ∈ *-Ring)
23	5, 6, 7, 8	ipcl 21966	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐵 , 𝐴) ∈ 𝐾)
24	1, 3, 4, 23	syl3anc 1399	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐵 , 𝐴) ∈ 𝐾)
25	18, 8, 10	srngmul 21136	. . . 4 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐶 ∈ 𝐾 ∧ (𝐵 , 𝐴) ∈ 𝐾) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
26	22, 2, 24, 25	syl3anc 1399	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐶 × (𝐵 , 𝐴))) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
27	13, 20, 26	3eqtr3d 2806	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)))
28	5, 6, 7, 18	ipcj 21967	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
29	1, 3, 4, 28	syl3anc 1399	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → ( ∗ ‘(𝐵 , 𝐴)) = (𝐴 , 𝐵))
30	29	oveq1d 7455	. 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (( ∗ ‘(𝐵 , 𝐴)) × ( ∗ ‘𝐶)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))
31	27, 30	eqtrd 2798	1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( ∗ ‘𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ‘cfv 6555 (class class class)co 7440 Basecbs 17413 ._rcmulr 17455 *_𝑟cstv 17456 Scalarcsca 17457 ·_𝑠 cvsca 17458 ·_𝑖cip 17459 *-Ringcsr 21122 LModclmod 21162 PreHilcphl 21957

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 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 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-tpos 8268 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-plusg 17467 df-mulr 17468 df-sca 17470 df-vsca 17471 df-ip 17472 df-0g 17638 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-mhm 19004 df-ghm 19454 df-mgp 20388 df-ur 20435 df-ring 20488 df-oppr 20594 df-rhm 20729 df-staf 21123 df-srng 21124 df-lmod 21164 df-lmhm 21324 df-lvec 21405 df-sra 21475 df-rgmod 21476 df-phl 21959

This theorem is used by: ipassr2 21980 cphassr 25560 tcphcphlem2 25584

W3C validator