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 2739 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | resvbas 31111 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐻)) |
4 | 3 | eleq2d 2819 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑒 ∈ (Base‘𝐺) ↔ 𝑒 ∈ (Base‘𝐻))) |
5 | eqid 2739 | . . . . . . . . 9 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
6 | 1, 5 | resvmulr 31114 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (.r‘𝐺) = (.r‘𝐻)) |
7 | 6 | oveqd 7190 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑒(.r‘𝐺)𝑥) = (𝑒(.r‘𝐻)𝑥)) |
8 | 7 | eqeq1d 2741 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑒(.r‘𝐺)𝑥) = 𝑥 ↔ (𝑒(.r‘𝐻)𝑥) = 𝑥)) |
9 | 6 | oveqd 7190 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑥(.r‘𝐺)𝑒) = (𝑥(.r‘𝐻)𝑒)) |
10 | 9 | eqeq1d 2741 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑥(.r‘𝐺)𝑒) = 𝑥 ↔ (𝑥(.r‘𝐻)𝑒) = 𝑥)) |
11 | 8, 10 | anbi12d 634 | . . . . 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 3305 | . . . 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 634 | . . 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 6324 | . 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 19376 | . 2 ⊢ 1 = (℩𝑒(𝑒 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)((𝑒(.r‘𝐺)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐺)𝑒) = 𝑥))) |
17 | eqid 2739 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
18 | eqid 2739 | . . 3 ⊢ (.r‘𝐻) = (.r‘𝐻) | |
19 | eqid 2739 | . . 3 ⊢ (1r‘𝐻) = (1r‘𝐻) | |
20 | 17, 18, 19 | dfur2 19376 | . 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 2799 | 1 ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ℩cio 6296 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 .rcmulr 16672 1rcur 19373 ↾v cresv 31103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-plusg 16684 df-mulr 16685 df-sca 16687 df-0g 16821 df-mgp 19362 df-ur 19374 df-resv 31104 |
This theorem is referenced by: (None) |
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