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Mirrors > Home > MPE Home > Th. List > opprlem | Structured version Visualization version GIF version |
Description: Lemma for opprbas 18452 and oppradd 18453. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
opprlem.2 | ⊢ 𝐸 = Slot 𝑁 |
opprlem.3 | ⊢ 𝑁 ∈ ℕ |
opprlem.4 | ⊢ 𝑁 < 3 |
Ref | Expression |
---|---|
opprlem | ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprlem.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | opprlem.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 15716 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 2 | nnrei 10906 | . . . . 5 ⊢ 𝑁 ∈ ℝ |
5 | opprlem.4 | . . . . 5 ⊢ 𝑁 < 3 | |
6 | 4, 5 | ltneii 10029 | . . . 4 ⊢ 𝑁 ≠ 3 |
7 | 1, 2 | ndxarg 15715 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
8 | mulrndx 15821 | . . . . 5 ⊢ (.r‘ndx) = 3 | |
9 | 7, 8 | neeq12i 2848 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (.r‘ndx) ↔ 𝑁 ≠ 3) |
10 | 6, 9 | mpbir 220 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
11 | 3, 10 | setsnid 15743 | . 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉)) |
12 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2610 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
14 | opprbas.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
15 | 12, 13, 14 | opprval 18447 | . . 3 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉) |
16 | 15 | fveq2i 6106 | . 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉)) |
17 | 11, 16 | eqtr4i 2635 | 1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 tpos ctpos 7238 < clt 9953 ℕcn 10897 3c3 10948 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 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 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
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-nel 2783 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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 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: opprbas 18452 oppradd 18453 |
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