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Mirrors > Home > MPE Home > Th. List > lmiclbs | Structured version Visualization version GIF version |
Description: Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
lmimlbs.j | ⊢ 𝐽 = (LBasis‘𝑆) |
lmimlbs.k | ⊢ 𝐾 = (LBasis‘𝑇) |
Ref | Expression |
---|---|
lmiclbs | ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 20942 | . . 3 ⊢ (𝑆 ≃𝑚 𝑇 ↔ (𝑆 LMIso 𝑇) ≠ ∅) | |
2 | n0 4342 | . . 3 ⊢ ((𝑆 LMIso 𝑇) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑆 LMIso 𝑇)) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (𝑆 ≃𝑚 𝑇 ↔ ∃𝑓 𝑓 ∈ (𝑆 LMIso 𝑇)) |
4 | n0 4342 | . . . 4 ⊢ (𝐽 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐽) | |
5 | lmimlbs.j | . . . . . . . 8 ⊢ 𝐽 = (LBasis‘𝑆) | |
6 | lmimlbs.k | . . . . . . . 8 ⊢ 𝐾 = (LBasis‘𝑇) | |
7 | 5, 6 | lmimlbs 21757 | . . . . . . 7 ⊢ ((𝑓 ∈ (𝑆 LMIso 𝑇) ∧ 𝑏 ∈ 𝐽) → (𝑓 “ 𝑏) ∈ 𝐾) |
8 | 7 | ne0d 4331 | . . . . . 6 ⊢ ((𝑓 ∈ (𝑆 LMIso 𝑇) ∧ 𝑏 ∈ 𝐽) → 𝐾 ≠ ∅) |
9 | 8 | ex 412 | . . . . 5 ⊢ (𝑓 ∈ (𝑆 LMIso 𝑇) → (𝑏 ∈ 𝐽 → 𝐾 ≠ ∅)) |
10 | 9 | exlimdv 1929 | . . . 4 ⊢ (𝑓 ∈ (𝑆 LMIso 𝑇) → (∃𝑏 𝑏 ∈ 𝐽 → 𝐾 ≠ ∅)) |
11 | 4, 10 | biimtrid 241 | . . 3 ⊢ (𝑓 ∈ (𝑆 LMIso 𝑇) → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) |
12 | 11 | exlimiv 1926 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑆 LMIso 𝑇) → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) |
13 | 3, 12 | sylbi 216 | 1 ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∅c0 4318 class class class wbr 5142 “ cima 5675 ‘cfv 6542 (class class class)co 7414 LMIso clmim 20894 ≃𝑚 clmic 20895 LBasisclbs 20948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-ghm 19159 df-mgp 20066 df-ur 20113 df-ring 20166 df-lmod 20734 df-lss 20805 df-lsp 20845 df-lmhm 20896 df-lmim 20897 df-lmic 20898 df-lbs 20949 df-lindf 21727 df-linds 21728 |
This theorem is referenced by: lmisfree 21763 |
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