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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmhmimasvsca | Structured version Visualization version GIF version |
Description: Value of the scalar product of the surjective image of a module. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
Ref | Expression |
---|---|
lmhmimasvsca.w | ⊢ 𝑊 = (𝐹 “s 𝑉) |
lmhmimasvsca.b | ⊢ 𝐵 = (Base‘𝑉) |
lmhmimasvsca.c | ⊢ 𝐶 = (Base‘𝑊) |
lmhmimasvsca.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
lmhmimasvsca.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
lmhmimasvsca.1 | ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶) |
lmhmimasvsca.f | ⊢ (𝜑 → 𝐹 ∈ (𝑉 LMHom 𝑊)) |
lmhmimasvsca.2 | ⊢ · = ( ·𝑠 ‘𝑉) |
lmhmimasvsca.3 | ⊢ × = ( ·𝑠 ‘𝑊) |
lmhmimasvsca.k | ⊢ 𝐾 = (Base‘(Scalar‘𝑉)) |
Ref | Expression |
---|---|
lmhmimasvsca | ⊢ (𝜑 → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmimasvsca.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
2 | lmhmimasvsca.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | lmhmimasvsca.w | . . . 4 ⊢ 𝑊 = (𝐹 “s 𝑉) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑊 = (𝐹 “s 𝑉)) |
5 | lmhmimasvsca.b | . . . 4 ⊢ 𝐵 = (Base‘𝑉) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑉)) |
7 | lmhmimasvsca.1 | . . 3 ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶) | |
8 | lmhmimasvsca.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑉 LMHom 𝑊)) | |
9 | lmhmlmod1 21050 | . . . 4 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑊) → 𝑉 ∈ LMod) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝜑 → 𝑉 ∈ LMod) |
11 | eqid 2734 | . . 3 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉) | |
12 | lmhmimasvsca.k | . . 3 ⊢ 𝐾 = (Base‘(Scalar‘𝑉)) | |
13 | lmhmimasvsca.2 | . . 3 ⊢ · = ( ·𝑠 ‘𝑉) | |
14 | lmhmimasvsca.3 | . . 3 ⊢ × = ( ·𝑠 ‘𝑊) | |
15 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → (𝐹‘𝑎) = (𝐹‘𝑞)) | |
16 | 15 | oveq2d 7461 | . . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → (𝑝 × (𝐹‘𝑎)) = (𝑝 × (𝐹‘𝑞))) |
17 | 8 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → 𝐹 ∈ (𝑉 LMHom 𝑊)) |
18 | simplr1 1215 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → 𝑝 ∈ 𝐾) | |
19 | simplr2 1216 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → 𝑎 ∈ 𝐵) | |
20 | 11, 12, 5, 13, 14 | lmhmlin 21052 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑉 LMHom 𝑊) ∧ 𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵) → (𝐹‘(𝑝 · 𝑎)) = (𝑝 × (𝐹‘𝑎))) |
21 | 17, 18, 19, 20 | syl3anc 1371 | . . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → (𝐹‘(𝑝 · 𝑎)) = (𝑝 × (𝐹‘𝑎))) |
22 | simplr3 1217 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → 𝑞 ∈ 𝐵) | |
23 | 11, 12, 5, 13, 14 | lmhmlin 21052 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑉 LMHom 𝑊) ∧ 𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝐵) → (𝐹‘(𝑝 · 𝑞)) = (𝑝 × (𝐹‘𝑞))) |
24 | 17, 18, 22, 23 | syl3anc 1371 | . . . . 5 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → (𝐹‘(𝑝 · 𝑞)) = (𝑝 × (𝐹‘𝑞))) |
25 | 16, 21, 24 | 3eqtr4d 2784 | . . . 4 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ (𝐹‘𝑎) = (𝐹‘𝑞)) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞))) |
26 | 25 | ex 412 | . . 3 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → ((𝐹‘𝑎) = (𝐹‘𝑞) → (𝐹‘(𝑝 · 𝑎)) = (𝐹‘(𝑝 · 𝑞)))) |
27 | 4, 6, 7, 10, 11, 12, 13, 14, 26 | imasvscaval 17593 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
28 | 1, 2, 27 | mpd3an23 1463 | 1 ⊢ (𝜑 → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 –onto→wfo 6570 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 Scalarcsca 17309 ·𝑠 cvsca 17310 “s cimas 17559 LModclmod 20875 LMHom clmhm 21036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-imas 17563 df-lmhm 21039 |
This theorem is referenced by: algextdeglem8 33707 |
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