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Theorem clmvscom 25067

Description: Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)

**Hypotheses**
Ref	Expression
clmvscl.v	⊢ 𝑉 = (Base‘𝑊)
clmvscl.f	⊢ 𝐹 = (Scalar‘𝑊)
clmvscl.s	⊢ · = ( ·_𝑠 ‘𝑊)
clmvscl.k	⊢ 𝐾 = (Base‘𝐹)

**Assertion**
Ref	Expression
clmvscom	⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))

**Proof of Theorem clmvscom**
Step	Hyp	Ref	Expression
1		ssel 3916	. . . . . . . 8 ⊢ (𝐾 ⊆ ℂ → (𝑄 ∈ 𝐾 → 𝑄 ∈ ℂ))
2		ssel 3916	. . . . . . . 8 ⊢ (𝐾 ⊆ ℂ → (𝑅 ∈ 𝐾 → 𝑅 ∈ ℂ))
3	1, 2	anim12d 610	. . . . . . 7 ⊢ (𝐾 ⊆ ℂ → ((𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) → (𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)))
4		clmvscl.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
5		clmvscl.k	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝐹)
6	4, 5	clmsscn 25056	. . . . . . 7 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
7	3, 6	syl11 33	. . . . . 6 ⊢ ((𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) → (𝑊 ∈ ℂMod → (𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)))
8	7	3adant3 1133	. . . . 5 ⊢ ((𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ ℂMod → (𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ)))
9	8	impcom 407	. . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ))
10		mulcom 11115	. . . 4 ⊢ ((𝑄 ∈ ℂ ∧ 𝑅 ∈ ℂ) → (𝑄 · 𝑅) = (𝑅 · 𝑄))
11	9, 10	syl 17	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · 𝑅) = (𝑅 · 𝑄))
12	11	oveq1d 7375	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 · 𝑅) · 𝑋) = ((𝑅 · 𝑄) · 𝑋))
13		clmvscl.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
14		clmvscl.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
15	13, 4, 14, 5	clmvsass 25066	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
16		3ancoma 1098	. . 3 ⊢ ((𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ↔ (𝑅 ∈ 𝐾 ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉))
17	13, 4, 14, 5	clmvsass 25066	. . 3 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑄) · 𝑋) = (𝑅 · (𝑄 · 𝑋)))
18	16, 17	sylan2b 595	. 2 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑄) · 𝑋) = (𝑅 · (𝑄 · 𝑋)))
19	12, 15, 18	3eqtr3d 2780	1 ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 · cmul 11034 Basecbs 17170 Scalarcsca 17214 ·_𝑠 cvsca 17215 ℂModcclm 25039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-mulf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-subrg 20538 df-lmod 20848 df-cnfld 21345 df-clm 25040

This theorem is referenced by: (None)

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