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Mirrors > Home > MPE Home > Th. List > ttglem | Structured version Visualization version GIF version |
Description: Lemma for ttgbas 27707, ttgvsca 27712 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
ttgval.n | ⊢ 𝐺 = (toTG‘𝐻) |
ttglem.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
ttglem.l | ⊢ (𝐸‘ndx) ≠ (LineG‘ndx) |
ttglem.i | ⊢ (𝐸‘ndx) ≠ (Itv‘ndx) |
Ref | Expression |
---|---|
ttglem | ⊢ (𝐸‘𝐻) = (𝐸‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ttglem.e | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | ttglem.i | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Itv‘ndx) | |
3 | 1, 2 | setsnid 17073 | . . . 4 ⊢ (𝐸‘𝐻) = (𝐸‘(𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉)) |
4 | ttglem.l | . . . . 5 ⊢ (𝐸‘ndx) ≠ (LineG‘ndx) | |
5 | 1, 4 | setsnid 17073 | . . . 4 ⊢ (𝐸‘(𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉)) = (𝐸‘((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ (𝑧 ∈ (𝑥(Itv‘𝐺)𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘𝐺)𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘𝐺)𝑧))})〉)) |
6 | 3, 5 | eqtri 2764 | . . 3 ⊢ (𝐸‘𝐻) = (𝐸‘((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ (𝑧 ∈ (𝑥(Itv‘𝐺)𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘𝐺)𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘𝐺)𝑧))})〉)) |
7 | ttgval.n | . . . . . 6 ⊢ 𝐺 = (toTG‘𝐻) | |
8 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
9 | eqid 2736 | . . . . . 6 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
10 | eqid 2736 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻) | |
11 | eqid 2736 | . . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
12 | 7, 8, 9, 10, 11 | ttgval 27703 | . . . . 5 ⊢ (𝐻 ∈ V → (𝐺 = ((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ (𝑧 ∈ (𝑥(Itv‘𝐺)𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘𝐺)𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘𝐺)𝑧))})〉) ∧ (Itv‘𝐺) = (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))}))) |
13 | 12 | simpld 495 | . . . 4 ⊢ (𝐻 ∈ V → 𝐺 = ((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ (𝑧 ∈ (𝑥(Itv‘𝐺)𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘𝐺)𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘𝐺)𝑧))})〉)) |
14 | 13 | fveq2d 6843 | . . 3 ⊢ (𝐻 ∈ V → (𝐸‘𝐺) = (𝐸‘((𝐻 sSet 〈(Itv‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝐻)𝑥) = (𝑘( ·𝑠 ‘𝐻)(𝑦(-g‘𝐻)𝑥))})〉) sSet 〈(LineG‘ndx), (𝑥 ∈ (Base‘𝐻), 𝑦 ∈ (Base‘𝐻) ↦ {𝑧 ∈ (Base‘𝐻) ∣ (𝑧 ∈ (𝑥(Itv‘𝐺)𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘𝐺)𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘𝐺)𝑧))})〉))) |
15 | 6, 14 | eqtr4id 2795 | . 2 ⊢ (𝐻 ∈ V → (𝐸‘𝐻) = (𝐸‘𝐺)) |
16 | 1 | str0 17053 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
17 | 16 | eqcomi 2745 | . . 3 ⊢ (𝐸‘∅) = ∅ |
18 | 17, 7 | fveqprc 17055 | . 2 ⊢ (¬ 𝐻 ∈ V → (𝐸‘𝐻) = (𝐸‘𝐺)) |
19 | 15, 18 | pm2.61i 182 | 1 ⊢ (𝐸‘𝐻) = (𝐸‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 {crab 3405 Vcvv 3443 ∅c0 4280 〈cop 4590 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 0cc0 11047 1c1 11048 [,]cicc 13259 sSet csts 17027 Slot cslot 17045 ndxcnx 17057 Basecbs 17075 ·𝑠 cvsca 17129 -gcsg 18742 Itvcitv 27261 LineGclng 27262 toTGcttg 27701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-dec 12615 df-sets 17028 df-slot 17046 df-ndx 17058 df-itv 27263 df-lng 27264 df-ttg 27702 |
This theorem is referenced by: ttgbas 27707 ttgplusg 27709 ttgvsca 27712 ttgds 27714 |
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