Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > resv1r | Structured version Visualization version GIF version |
Description: 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
Ref | Expression |
---|---|
resvbas.1 | ⊢ 𝐻 = (𝐺 ↾v 𝐴) |
resv1r.2 | ⊢ 1 = (1r‘𝐺) |
Ref | Expression |
---|---|
resv1r | ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resvbas.1 | . . . . . 6 ⊢ 𝐻 = (𝐺 ↾v 𝐴) | |
2 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | resvbas 30905 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐻)) |
4 | 3 | eleq2d 2898 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑒 ∈ (Base‘𝐺) ↔ 𝑒 ∈ (Base‘𝐻))) |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
6 | 1, 5 | resvmulr 30908 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (.r‘𝐺) = (.r‘𝐻)) |
7 | 6 | oveqd 7172 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑒(.r‘𝐺)𝑥) = (𝑒(.r‘𝐻)𝑥)) |
8 | 7 | eqeq1d 2823 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑒(.r‘𝐺)𝑥) = 𝑥 ↔ (𝑒(.r‘𝐻)𝑥) = 𝑥)) |
9 | 6 | oveqd 7172 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥(.r‘𝐺)𝑒) = (𝑥(.r‘𝐻)𝑒)) |
10 | 9 | eqeq1d 2823 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥(.r‘𝐺)𝑒) = 𝑥 ↔ (𝑥(.r‘𝐻)𝑒) = 𝑥)) |
11 | 8, 10 | anbi12d 632 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥) ↔ ((𝑒(.r‘𝐻)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐻)𝑒) = 𝑥))) |
12 | 3, 11 | raleqbidv 3401 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ (Base‘𝐺)((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑒(.r‘𝐻)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐻)𝑒) = 𝑥))) |
13 | 4, 12 | anbi12d 632 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑒 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥)) ↔ (𝑒 ∈ (Base‘𝐻) ∧ ∀𝑥 ∈ (Base‘𝐻)((𝑒(.r‘𝐻)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐻)𝑒) = 𝑥)))) |
14 | 13 | iotabidv 6338 | . 2 ⊢ (𝐴 ∈ 𝑉 → (℩𝑒(𝑒 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥))) = (℩𝑒(𝑒 ∈ (Base‘𝐻) ∧ ∀𝑥 ∈ (Base‘𝐻)((𝑒(.r‘𝐻)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐻)𝑒) = 𝑥)))) |
15 | resv1r.2 | . . 3 ⊢ 1 = (1r‘𝐺) | |
16 | 2, 5, 15 | dfur2 19253 | . 2 ⊢ 1 = (℩𝑒(𝑒 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥))) |
17 | eqid 2821 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
18 | eqid 2821 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
19 | eqid 2821 | . . 3 ⊢ (1r‘𝐻) = (1r‘𝐻) | |
20 | 17, 18, 19 | dfur2 19253 | . 2 ⊢ (1r‘𝐻) = (℩𝑒(𝑒 ∈ (Base‘𝐻) ∧ ∀𝑥 ∈ (Base‘𝐻)((𝑒(.r‘𝐻)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐻)𝑒) = 𝑥))) |
21 | 14, 16, 20 | 3eqtr4g 2881 | 1 ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ℩cio 6311 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 .rcmulr 16565 1rcur 19250 ↾v cresv 30897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-mulr 16578 df-sca 16580 df-0g 16714 df-mgp 19239 df-ur 19251 df-resv 30898 |
This theorem is referenced by: (None) |
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