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Mirrors > Home > MPE Home > Th. List > Mathboxes > dimpropd | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), they have the same dimension. (Contributed by Thierry Arnoux, 18-May-2023.) |
Ref | Expression |
---|---|
dimpropd.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
dimpropd.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
dimpropd.w | ⊢ (𝜑 → 𝐵 ⊆ 𝑊) |
dimpropd.p | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
dimpropd.s1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) |
dimpropd.s2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) |
dimpropd.f | ⊢ 𝐹 = (Scalar‘𝐾) |
dimpropd.g | ⊢ 𝐺 = (Scalar‘𝐿) |
dimpropd.p1 | ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) |
dimpropd.p2 | ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) |
dimpropd.a | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
dimpropd.v1 | ⊢ (𝜑 → 𝐾 ∈ LVec) |
dimpropd.v2 | ⊢ (𝜑 → 𝐿 ∈ LVec) |
Ref | Expression |
---|---|
dimpropd | ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dimpropd.v1 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ LVec) | |
2 | eqid 2736 | . . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾) | |
3 | 2 | lbsex 21159 | . . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → (LBasis‘𝐾) ≠ ∅) |
5 | n0 4352 | . . 3 ⊢ ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐾)) | |
6 | 4, 5 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐾)) |
7 | 2 | dimval 33638 | . . . 4 ⊢ ((𝐾 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑥)) |
8 | 1, 7 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑥)) |
9 | dimpropd.v2 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ LVec) | |
10 | dimpropd.b1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
11 | dimpropd.b2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
12 | dimpropd.w | . . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑊) | |
13 | dimpropd.p | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
14 | dimpropd.s1 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) | |
15 | dimpropd.s2 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) | |
16 | dimpropd.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝐾) | |
17 | dimpropd.g | . . . . . . 7 ⊢ 𝐺 = (Scalar‘𝐿) | |
18 | dimpropd.p1 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) | |
19 | dimpropd.p2 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) | |
20 | dimpropd.a | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) | |
21 | 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 9 | lbspropd 21090 | . . . . . 6 ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿)) |
22 | 21 | eleq2d 2826 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (LBasis‘𝐾) ↔ 𝑥 ∈ (LBasis‘𝐿))) |
23 | 22 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → 𝑥 ∈ (LBasis‘𝐿)) |
24 | eqid 2736 | . . . . 5 ⊢ (LBasis‘𝐿) = (LBasis‘𝐿) | |
25 | 24 | dimval 33638 | . . . 4 ⊢ ((𝐿 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐿)) → (dim‘𝐿) = (♯‘𝑥)) |
26 | 9, 23, 25 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐿) = (♯‘𝑥)) |
27 | 8, 26 | eqtr4d 2779 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (dim‘𝐿)) |
28 | 6, 27 | exlimddv 1935 | 1 ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 ‘cfv 6559 (class class class)co 7429 ♯chash 14365 Basecbs 17243 +gcplusg 17293 Scalarcsca 17296 ·𝑠 cvsca 17297 LBasisclbs 21065 LVecclvec 21093 dimcldim 33636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-reg 9628 ax-inf2 9677 ax-ac2 10499 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-rpss 7739 df-om 7884 df-1st 8010 df-2nd 8011 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-oadd 8506 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-oi 9546 df-r1 9800 df-rank 9801 df-dju 9937 df-card 9975 df-acn 9978 df-ac 10152 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-tset 17312 df-ple 17313 df-ocomp 17314 df-0g 17482 df-mre 17625 df-mrc 17626 df-mri 17627 df-acs 17628 df-proset 18336 df-drs 18337 df-poset 18355 df-ipo 18569 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-subg 19137 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-ring 20228 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-drng 20723 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lbs 21066 df-lvec 21094 df-dim 33637 |
This theorem is referenced by: tngdim 33651 matdim 33653 algextdeglem8 33746 |
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