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Mirrors > Home > MPE Home > Th. List > matplusg | Structured version Visualization version Unicode version |
Description: The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matbas.a |
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matbas.g |
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Ref | Expression |
---|---|
matplusg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matbas.a |
. . . 4
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2 | matbas.g |
. . . 4
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3 | eqid 2450 |
. . . 4
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4 | 1, 2, 3 | matval 19429 |
. . 3
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5 | 4 | fveq2d 5867 |
. 2
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6 | plusgid 15218 |
. . 3
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7 | plusgndx 15217 |
. . . 4
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8 | 2re 10676 |
. . . . . 6
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9 | 2lt3 10774 |
. . . . . 6
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10 | 8, 9 | ltneii 9744 |
. . . . 5
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11 | mulrndx 15235 |
. . . . 5
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12 | 10, 11 | neeqtrri 2696 |
. . . 4
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13 | 7, 12 | eqnetri 2693 |
. . 3
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14 | 6, 13 | setsnid 15158 |
. 2
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15 | 5, 14 | syl6reqr 2503 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-ot 3976 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-nn 10607 df-2 10665 df-3 10666 df-ndx 15117 df-slot 15118 df-sets 15120 df-plusg 15196 df-mulr 15197 df-mat 19426 |
This theorem is referenced by: mat0 19435 matinvg 19436 matplusg2 19445 matlmod 19447 matsubg 19450 matgsum 19455 |
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