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Mirrors > Home > MPE Home > Th. List > 2strstr1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of 2strstr1 17202 as of 27-Oct-2024. A constructed two-slot structure. Version of 2strstr 17199 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2str1.g | ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩} |
2str1.b | ⊢ (Base‘ndx) < 𝑁 |
2str1.n | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
2strstr1OLD | ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2str1.g | . . . 4 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩} | |
2 | eqid 2725 | . . . . . . . 8 ⊢ Slot 𝑁 = Slot 𝑁 | |
3 | 2str1.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxarg 17162 | . . . . . . 7 ⊢ (Slot 𝑁‘ndx) = 𝑁 |
5 | 4 | eqcomi 2734 | . . . . . 6 ⊢ 𝑁 = (Slot 𝑁‘ndx) |
6 | 5 | opeq1i 4872 | . . . . 5 ⊢ ⟨𝑁, + ⟩ = ⟨(Slot 𝑁‘ndx), + ⟩ |
7 | 6 | preq2i 4737 | . . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Slot 𝑁‘ndx), + ⟩} |
8 | 1, 7 | eqtri 2753 | . . 3 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Slot 𝑁‘ndx), + ⟩} |
9 | basendx 17186 | . . . 4 ⊢ (Base‘ndx) = 1 | |
10 | 2str1.b | . . . 4 ⊢ (Base‘ndx) < 𝑁 | |
11 | 9, 10 | eqbrtrri 5166 | . . 3 ⊢ 1 < 𝑁 |
12 | 8, 2, 11, 3 | 2strstr 17199 | . 2 ⊢ 𝐺 Struct ⟨1, 𝑁⟩ |
13 | 9 | opeq1i 4872 | . 2 ⊢ ⟨(Base‘ndx), 𝑁⟩ = ⟨1, 𝑁⟩ |
14 | 12, 13 | breqtrri 5170 | 1 ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cpr 4626 ⟨cop 4630 class class class wbr 5143 ‘cfv 6542 1c1 11137 < clt 11276 ℕcn 12240 Struct cstr 17112 Slot cslot 17147 ndxcnx 17159 Basecbs 17177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 |
This theorem is referenced by: (None) |
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