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Mirrors > Home > MPE Home > Th. List > ttglem | Structured version Visualization version GIF version |
Description: Lemma for ttgbas 28806, ttgvsca 28811 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 17211 | . . . 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 17211 | . . . 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 2754 | . . 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 2726 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
9 | eqid 2726 | . . . . . 6 ⊢ (-g‘𝐻) = (-g‘𝐻) | |
10 | eqid 2726 | . . . . . 6 ⊢ ( ·𝑠 ‘𝐻) = ( ·𝑠 ‘𝐻) | |
11 | eqid 2726 | . . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
12 | 7, 8, 9, 10, 11 | ttgval 28802 | . . . . 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 493 | . . . 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 6905 | . . 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 2785 | . 2 ⊢ (𝐻 ∈ V → (𝐸‘𝐻) = (𝐸‘𝐺)) |
16 | 1 | str0 17191 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
17 | 16 | eqcomi 2735 | . . 3 ⊢ (𝐸‘∅) = ∅ |
18 | 17, 7 | fveqprc 17193 | . 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 1083 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 {crab 3419 Vcvv 3462 ∅c0 4325 〈cop 4639 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 0cc0 11158 1c1 11159 [,]cicc 13381 sSet csts 17165 Slot cslot 17183 ndxcnx 17195 Basecbs 17213 ·𝑠 cvsca 17270 -gcsg 18930 Itvcitv 28360 LineGclng 28361 toTGcttg 28800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12730 df-sets 17166 df-slot 17184 df-ndx 17196 df-itv 28362 df-lng 28363 df-ttg 28801 |
This theorem is referenced by: ttgbas 28806 ttgplusg 28808 ttgvsca 28811 ttgds 28813 |
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