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Mirrors > Home > MPE Home > Th. List > opprlemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opprlem 20356 as of 6-Nov-2024. Lemma for opprbas 20358 and oppradd 20360. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
opprlemOLD.2 | ⊢ 𝐸 = Slot 𝑁 |
opprlemOLD.3 | ⊢ 𝑁 ∈ ℕ |
opprlemOLD.4 | ⊢ 𝑁 < 3 |
Ref | Expression |
---|---|
opprlemOLD | ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprlemOLD.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | opprlemOLD.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 17231 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 2 | nnrei 12273 | . . . . 5 ⊢ 𝑁 ∈ ℝ |
5 | opprlemOLD.4 | . . . . 5 ⊢ 𝑁 < 3 | |
6 | 4, 5 | ltneii 11372 | . . . 4 ⊢ 𝑁 ≠ 3 |
7 | 1, 2 | ndxarg 17230 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
8 | mulrndx 17339 | . . . . 5 ⊢ (.r‘ndx) = 3 | |
9 | 7, 8 | neeq12i 3005 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (.r‘ndx) ↔ 𝑁 ≠ 3) |
10 | 6, 9 | mpbir 231 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) |
11 | 3, 10 | setsnid 17243 | . 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉)) |
12 | eqid 2735 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | eqid 2735 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
14 | opprbas.1 | . . . 4 ⊢ 𝑂 = (oppr‘𝑅) | |
15 | 12, 13, 14 | opprval 20352 | . . 3 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉) |
16 | 15 | fveq2i 6910 | . 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉)) |
17 | 11, 16 | eqtr4i 2766 | 1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ≠ wne 2938 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 tpos ctpos 8249 < clt 11293 ℕcn 12264 3c3 12320 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 Basecbs 17245 .rcmulr 17299 opprcoppr 20350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-mulr 17312 df-oppr 20351 |
This theorem is referenced by: opprbasOLD 20359 oppraddOLD 20361 |
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