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Mirrors > Home > MPE Home > Th. List > opprmulfval | Structured version Visualization version GIF version |
Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
opprmulfval.4 | ⊢ ∙ = (.r‘𝑂) |
Ref | Expression |
---|---|
opprmulfval | ⊢ ∙ = tpos · |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprmulfval.4 | . 2 ⊢ ∙ = (.r‘𝑂) | |
2 | opprval.2 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
3 | fvex 6113 | . . . . . . 7 ⊢ (.r‘𝑅) ∈ V | |
4 | 2, 3 | eqeltri 2684 | . . . . . 6 ⊢ · ∈ V |
5 | 4 | tposex 7273 | . . . . 5 ⊢ tpos · ∈ V |
6 | mulrid 15822 | . . . . . 6 ⊢ .r = Slot (.r‘ndx) | |
7 | 6 | setsid 15742 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
8 | 5, 7 | mpan2 703 | . . . 4 ⊢ (𝑅 ∈ V → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
9 | opprval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
10 | opprval.3 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
11 | 9, 2, 10 | opprval 18447 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
12 | 11 | fveq2i 6106 | . . . 4 ⊢ (.r‘𝑂) = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
13 | 8, 12 | syl6reqr 2663 | . . 3 ⊢ (𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
14 | tpos0 7269 | . . . . 5 ⊢ tpos ∅ = ∅ | |
15 | 6 | str0 15739 | . . . . 5 ⊢ ∅ = (.r‘∅) |
16 | 14, 15 | eqtr2i 2633 | . . . 4 ⊢ (.r‘∅) = tpos ∅ |
17 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
18 | 10, 17 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
19 | 18 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = (.r‘∅)) |
20 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑅) = ∅) | |
21 | 2, 20 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → · = ∅) |
22 | 21 | tposeqd 7242 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos · = tpos ∅) |
23 | 16, 19, 22 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
24 | 13, 23 | pm2.61i 175 | . 2 ⊢ (.r‘𝑂) = tpos · |
25 | 1, 24 | eqtri 2632 | 1 ⊢ ∙ = tpos · |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 〈cop 4131 ‘cfv 5804 (class class class)co 6549 tpos ctpos 7238 ndxcnx 15692 sSet csts 15693 Basecbs 15695 .rcmulr 15769 opprcoppr 18445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-sets 15701 df-mulr 15782 df-oppr 18446 |
This theorem is referenced by: opprmul 18449 |
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