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Mirrors > Home > MPE Home > Th. List > 2strstr1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of 2strstr1 17178 as of 27-Oct-2024. A constructed two-slot structure. Version of 2strstr 17175 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 2726 | . . . . . . . 8 ⊢ Slot 𝑁 = Slot 𝑁 | |
3 | 2str1.n | . . . . . . . 8 ⊢ 𝑁 ∈ ℕ | |
4 | 2, 3 | ndxarg 17138 | . . . . . . 7 ⊢ (Slot 𝑁‘ndx) = 𝑁 |
5 | 4 | eqcomi 2735 | . . . . . 6 ⊢ 𝑁 = (Slot 𝑁‘ndx) |
6 | 5 | opeq1i 4871 | . . . . 5 ⊢ ⟨𝑁, + ⟩ = ⟨(Slot 𝑁‘ndx), + ⟩ |
7 | 6 | preq2i 4736 | . . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨𝑁, + ⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Slot 𝑁‘ndx), + ⟩} |
8 | 1, 7 | eqtri 2754 | . . 3 ⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Slot 𝑁‘ndx), + ⟩} |
9 | basendx 17162 | . . . 4 ⊢ (Base‘ndx) = 1 | |
10 | 2str1.b | . . . 4 ⊢ (Base‘ndx) < 𝑁 | |
11 | 9, 10 | eqbrtrri 5164 | . . 3 ⊢ 1 < 𝑁 |
12 | 8, 2, 11, 3 | 2strstr 17175 | . 2 ⊢ 𝐺 Struct ⟨1, 𝑁⟩ |
13 | 9 | opeq1i 4871 | . 2 ⊢ ⟨(Base‘ndx), 𝑁⟩ = ⟨1, 𝑁⟩ |
14 | 12, 13 | breqtrri 5168 | 1 ⊢ 𝐺 Struct ⟨(Base‘ndx), 𝑁⟩ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cpr 4625 ⟨cop 4629 class class class wbr 5141 ‘cfv 6537 1c1 11113 < clt 11252 ℕcn 12216 Struct cstr 17088 Slot cslot 17123 ndxcnx 17135 Basecbs 17153 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 |
This theorem is referenced by: (None) |
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