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| Mirrors > Home > MPE Home > Th. List > ttglem | Structured version Visualization version GIF version | ||
| Description: Lemma for ttgbas 28887, ttgvsca 28892 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 17245 | . . . 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 17245 | . . . 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 2765 | . . 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 2737 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻) | |
| 11 | eqid 2737 | . . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
| 12 | 7, 8, 9, 10, 11 | ttgval 28883 | . . . . 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 494 | . . . 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 6910 | . . 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 2796 | . 2 ⊢ (𝐻 ∈ V → (𝐸‘𝐻) = (𝐸‘𝐺)) |
| 16 | 1 | str0 17226 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
| 17 | 16 | eqcomi 2746 | . . 3 ⊢ (𝐸‘∅) = ∅ |
| 18 | 17, 7 | fveqprc 17228 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 0cc0 11155 1c1 11156 [,]cicc 13390 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 Basecbs 17247 ·𝑠 cvsca 17301 -gcsg 18953 Itvcitv 28441 LineGclng 28442 toTGcttg 28881 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-itv 28443 df-lng 28444 df-ttg 28882 |
| This theorem is referenced by: ttgbas 28887 ttgplusg 28889 ttgvsca 28892 ttgds 28894 |
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