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Mirrors > Home > ILE Home > Th. List > lspsnsub | GIF version |
Description: Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.) |
Ref | Expression |
---|---|
lspsnsub.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnsub.s | ⊢ − = (-g‘𝑊) |
lspsnsub.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lspsnsub.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnsub.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspsnsub.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
lspsnsub | ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnsub.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
2 | lspsnsub.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | lspsnsub.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
4 | lspsnsub.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
5 | lspsnsub.s | . . . . 5 ⊢ − = (-g‘𝑊) | |
6 | 4, 5 | lmodvsubcl 13578 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉) |
7 | 1, 2, 3, 6 | syl3anc 1248 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉) |
8 | eqid 2187 | . . . 4 ⊢ (invg‘𝑊) = (invg‘𝑊) | |
9 | lspsnsub.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
10 | 4, 8, 9 | lspsnneg 13666 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{((invg‘𝑊)‘(𝑋 − 𝑌))}) = (𝑁‘{(𝑋 − 𝑌)})) |
11 | 1, 7, 10 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘(𝑋 − 𝑌))}) = (𝑁‘{(𝑋 − 𝑌)})) |
12 | lmodgrp 13540 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
13 | 1, 12 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp) |
14 | 4, 5, 8 | grpinvsub 12987 | . . . . 5 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((invg‘𝑊)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
15 | 13, 2, 3, 14 | syl3anc 1248 | . . . 4 ⊢ (𝜑 → ((invg‘𝑊)‘(𝑋 − 𝑌)) = (𝑌 − 𝑋)) |
16 | 15 | sneqd 3617 | . . 3 ⊢ (𝜑 → {((invg‘𝑊)‘(𝑋 − 𝑌))} = {(𝑌 − 𝑋)}) |
17 | 16 | fveq2d 5531 | . 2 ⊢ (𝜑 → (𝑁‘{((invg‘𝑊)‘(𝑋 − 𝑌))}) = (𝑁‘{(𝑌 − 𝑋)})) |
18 | 11, 17 | eqtr3d 2222 | 1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 {csn 3604 ‘cfv 5228 (class class class)co 5888 Basecbs 12476 Grpcgrp 12906 invgcminusg 12907 -gcsg 12908 LModclmod 13533 LSpanclspn 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-ndx 12479 df-slot 12480 df-base 12482 df-sets 12483 df-plusg 12564 df-mulr 12565 df-sca 12567 df-vsca 12568 df-0g 12725 df-mgm 12794 df-sgrp 12827 df-mnd 12840 df-grp 12909 df-minusg 12910 df-sbg 12911 df-mgp 13230 df-ur 13269 df-ring 13307 df-lmod 13535 df-lssm 13599 df-lsp 13633 |
This theorem is referenced by: (None) |
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