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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 39828 | . . . 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 39822 | . . 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 19643 | . . . 4 ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) |
13 | 10, 11, 12 | mpbir2and 713 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) |
14 | 1, 2, 5, 4, 9, 13 | drnginvrcld 39830 | . 2 ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) ∈ 𝐵) |
15 | 1, 2, 5, 4, 8, 11 | drnginvrcld 39830 | . . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ 𝐵) |
16 | 1, 2, 5, 4, 7, 10 | drnginvrcld 39830 | . . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵) |
17 | 1, 3, 6, 15, 16 | ringcld 39822 | . 2 ⊢ (𝜑 → ((𝐼‘𝑌) · (𝐼‘𝑋)) ∈ 𝐵) |
18 | eqid 2738 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
19 | 1, 2, 3, 18, 5, 4, 7, 10 | drnginvrld 39832 | . . . . . . . 8 ⊢ (𝜑 → ((𝐼‘𝑋) · 𝑋) = (1r‘𝑅)) |
20 | 19 | oveq1d 7185 | . . . . . . 7 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = ((1r‘𝑅) · 𝑌)) |
21 | 1, 3, 18, 6, 8 | ringlidmd 39824 | . . . . . . 7 ⊢ (𝜑 → ((1r‘𝑅) · 𝑌) = 𝑌) |
22 | 20, 21 | eqtrd 2773 | . . . . . 6 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = 𝑌) |
23 | 22 | oveq2d 7186 | . . . . 5 ⊢ (𝜑 → ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌)) = ((𝐼‘𝑌) · 𝑌)) |
24 | 23 | eqcomd 2744 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · 𝑌) = ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌))) |
25 | 1, 2, 3, 18, 5, 4, 8, 11 | drnginvrld 39832 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · 𝑌) = (1r‘𝑅)) |
26 | 1, 3, 6, 16, 7, 8 | ringassd 39823 | . . . . 5 ⊢ (𝜑 → (((𝐼‘𝑋) · 𝑋) · 𝑌) = ((𝐼‘𝑋) · (𝑋 · 𝑌))) |
27 | 26 | oveq2d 7186 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑌) · (((𝐼‘𝑋) · 𝑋) · 𝑌)) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
28 | 24, 25, 27 | 3eqtr3d 2781 | . . 3 ⊢ (𝜑 → (1r‘𝑅) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
29 | 1, 2, 3, 18, 5, 4, 9, 13 | drnginvrld 39832 | . . 3 ⊢ (𝜑 → ((𝐼‘(𝑋 · 𝑌)) · (𝑋 · 𝑌)) = (1r‘𝑅)) |
30 | 1, 3, 6, 15, 16, 9 | ringassd 39823 | . . 3 ⊢ (𝜑 → (((𝐼‘𝑌) · (𝐼‘𝑋)) · (𝑋 · 𝑌)) = ((𝐼‘𝑌) · ((𝐼‘𝑋) · (𝑋 · 𝑌)))) |
31 | 28, 29, 30 | 3eqtr4d 2783 | . 2 ⊢ (𝜑 → ((𝐼‘(𝑋 · 𝑌)) · (𝑋 · 𝑌)) = (((𝐼‘𝑌) · (𝐼‘𝑋)) · (𝑋 · 𝑌))) |
32 | 1, 2, 3, 4, 14, 17, 9, 13, 31 | drngmulcan2ad 39835 | 1 ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 .rcmulr 16669 0gc0g 16816 1rcur 19370 invrcinvr 19543 DivRingcdr 19621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-drng 19623 |
This theorem is referenced by: prjspner1 40040 |
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