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Mirrors > Home > MPE Home > Th. List > Mathboxes > drnginvmuld | Structured version Visualization version GIF version |
Description: Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.) |
Ref | Expression |
---|---|
drnginvmuld.b | ⊢ 𝐵 = (Base‘𝑅) |
drnginvmuld.z | ⊢ 0 = (0g‘𝑅) |
drnginvmuld.t | ⊢ · = (.r‘𝑅) |
drnginvmuld.i | ⊢ 𝐼 = (invr‘𝑅) |
drnginvmuld.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
drnginvmuld.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
drnginvmuld.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
drnginvmuld.1 | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
drnginvmuld.2 | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
Ref | Expression |
---|---|
drnginvmuld | ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drnginvmuld.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | drnginvmuld.z | . 2 ⊢ 0 = (0g‘𝑅) | |
3 | drnginvmuld.t | . 2 ⊢ · = (.r‘𝑅) | |
4 | drnginvmuld.r | . 2 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
5 | drnginvmuld.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
6 | 4 | drngringd 39782 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
7 | drnginvmuld.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | drnginvmuld.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 1, 3, 6, 7, 8 | ringcld 39776 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
10 | drnginvmuld.1 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
11 | drnginvmuld.2 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
12 | 1, 2, 3, 4, 7, 8 | drngmulne0 19597 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) |
13 | 10, 11, 12 | mpbir2and 712 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) |
14 | 1, 2, 5, 4, 9, 13 | drnginvrcld 39784 | . 2 ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) ∈ 𝐵) |
15 | 1, 2, 5, 4, 8, 11 | drnginvrcld 39784 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ 𝐵) |
16 | 1, 2, 5, 4, 7, 10 | drnginvrcld 39784 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) |
17 | 1, 3, 6, 15, 16 | ringcld 39776 | . 2 ⊢ (𝜑 → ((𝐼‘𝑌) · (𝐼‘𝑋)) ∈ 𝐵) |
18 | eqid 2758 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
19 | 1, 2, 3, 18, 5, 4, 7, 10 | drnginvrld 39786 | . . . . . . . 8 ⊢ (𝜑 → ((𝐼‘𝑋) · 𝑋) = (1r‘𝑅)) |
20 | 19 | oveq1d 7170 | . . . . . . 7 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = ((1r‘𝑅) · 𝑌)) |
21 | 1, 3, 18, 6, 8 | ringlidmd 39778 | . . . . . . 7 ⊢ (𝜑 → ((1r‘𝑅) · 𝑌) = 𝑌) |
22 | 20, 21 | eqtrd 2793 | . . . . . 6 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = 𝑌) |
23 | 22 | oveq2d 7171 | . . . . 5 ⊢ (𝜑 → ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌)) = ((𝐼‘𝑌) · 𝑌)) |
24 | 23 | eqcomd 2764 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · 𝑌) = ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌))) |
25 | 1, 2, 3, 18, 5, 4, 8, 11 | drnginvrld 39786 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · 𝑌) = (1r‘𝑅)) |
26 | 1, 3, 6, 16, 7, 8 | ringassd 39777 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = ((𝐼‘𝑋) · (𝑋 · 𝑌))) |
27 | 26 | oveq2d 7171 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌)) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
28 | 24, 25, 27 | 3eqtr3d 2801 | . . 3 ⊢ (𝜑 → (1r‘𝑅) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
29 | 1, 2, 3, 18, 5, 4, 9, 13 | drnginvrld 39786 | . . 3 ⊢ (𝜑 → ((𝐼‘(𝑋 · 𝑌)) · (𝑋 · 𝑌)) = (1r‘𝑅)) |
30 | 1, 3, 6, 15, 16, 9 | ringassd 39777 | . . 3 ⊢ (𝜑 → (((𝐼‘𝑌) · (𝐼‘𝑋)) · (𝑋 · 𝑌)) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
31 | 28, 29, 30 | 3eqtr4d 2803 | . 2 ⊢ (𝜑 → ((𝐼‘(𝑋 · 𝑌)) · (𝑋 · 𝑌)) = (((𝐼‘𝑌) · (𝐼‘𝑋)) · (𝑋 · 𝑌))) |
32 | 1, 2, 3, 4, 14, 17, 9, 13, 31 | drngmulcan2ad 39789 | 1 ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ‘cfv 6339 (class class class)co 7155 Basecbs 16546 .rcmulr 16629 0gc0g 16776 1rcur 19324 invrcinvr 19497 DivRingcdr 19575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7585 df-tpos 7907 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-er 8304 df-en 8533 df-dom 8534 df-sdom 8535 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-nn 11680 df-2 11742 df-3 11743 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-0g 16778 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-grp 18177 df-minusg 18178 df-mgp 19313 df-ur 19325 df-ring 19372 df-oppr 19449 df-dvdsr 19467 df-unit 19468 df-invr 19498 df-drng 19577 |
This theorem is referenced by: prjspner1 39988 |
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