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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 2821 | . . . . 5 ⊢ (LBasis‘𝐾) = (LBasis‘𝐾) | |
3 | 2 | lbsex 19931 | . . . 4 ⊢ (𝐾 ∈ LVec → (LBasis‘𝐾) ≠ ∅) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → (LBasis‘𝐾) ≠ ∅) |
5 | n0 4309 | . . 3 ⊢ ((LBasis‘𝐾) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (LBasis‘𝐾)) | |
6 | 4, 5 | sylib 220 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ (LBasis‘𝐾)) |
7 | 2 | dimval 30996 | . . . 4 ⊢ ((𝐾 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (♯‘𝑥)) |
8 | 1, 7 | sylan 582 | . . 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 | 1 | elexd 3514 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ V) |
22 | 9 | elexd 3514 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ V) |
23 | 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | lbspropd 19865 | . . . . . 6 ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿)) |
24 | 23 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (LBasis‘𝐾) ↔ 𝑥 ∈ (LBasis‘𝐿))) |
25 | 24 | biimpa 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → 𝑥 ∈ (LBasis‘𝐿)) |
26 | eqid 2821 | . . . . 5 ⊢ (LBasis‘𝐿) = (LBasis‘𝐿) | |
27 | 26 | dimval 30996 | . . . 4 ⊢ ((𝐿 ∈ LVec ∧ 𝑥 ∈ (LBasis‘𝐿)) → (dim‘𝐿) = (♯‘𝑥)) |
28 | 9, 25, 27 | syl2an2r 683 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐿) = (♯‘𝑥)) |
29 | 8, 28 | eqtr4d 2859 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (LBasis‘𝐾)) → (dim‘𝐾) = (dim‘𝐿)) |
30 | 6, 29 | exlimddv 1932 | 1 ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 ♯chash 13684 Basecbs 16477 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 LBasisclbs 19840 LVecclvec 19868 dimcldim 30994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-reg 9050 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rpss 7443 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-r1 9187 df-rank 9188 df-dju 9324 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ple 16579 df-ocomp 16580 df-0g 16709 df-mre 16851 df-mrc 16852 df-mri 16853 df-acs 16854 df-proset 17532 df-drs 17533 df-poset 17550 df-ipo 17756 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lbs 19841 df-lvec 19869 df-dim 30995 |
This theorem is referenced by: tngdim 31006 matdim 31008 |
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