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Mirrors > Home > MPE Home > Th. List > lspsnsubn0 | Structured version Visualization version GIF version |
Description: Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.) |
Ref | Expression |
---|---|
lspsnsubn0.v | ⊢ 𝑉 = (Base‘𝑊) |
lspsnsubn0.o | ⊢ 0 = (0g‘𝑊) |
lspsnsubn0.m | ⊢ − = (-g‘𝑊) |
lspsnsubn0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lspsnsubn0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lspsnsubn0.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
lspsnsubn0.e | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lspsnsubn0 | ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnsubn0.e | . 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
2 | lspsnsubn0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lspsnsubn0.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | lspsnsubn0.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
5 | lspsnsubn0.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
6 | lspsnsubn0.o | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
7 | lspsnsubn0.m | . . . . . 6 ⊢ − = (-g‘𝑊) | |
8 | 5, 6, 7 | lmodsubeq0 20288 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
9 | 2, 3, 4, 8 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 ↔ 𝑋 = 𝑌)) |
10 | sneq 4583 | . . . . 5 ⊢ (𝑋 = 𝑌 → {𝑋} = {𝑌}) | |
11 | 10 | fveq2d 6829 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
12 | 9, 11 | syl6bi 252 | . . 3 ⊢ (𝜑 → ((𝑋 − 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) |
13 | 12 | necon3d 2961 | . 2 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → (𝑋 − 𝑌) ≠ 0 )) |
14 | 1, 13 | mpd 15 | 1 ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {csn 4573 ‘cfv 6479 (class class class)co 7337 Basecbs 17009 0gc0g 17247 -gcsg 18675 LModclmod 20229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-minusg 18677 df-sbg 18678 df-lmod 20231 |
This theorem is referenced by: mapdpglem4N 39952 |
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