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Mirrors > Home > MPE Home > Th. List > mulgfvi | Structured version Visualization version GIF version |
Description: The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
mulgfvi.t | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulgfvi | ⊢ · = (.g‘( I ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgfvi.t | . 2 ⊢ · = (.g‘𝐺) | |
2 | fvi 6165 | . . . . 5 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | eqcomd 2616 | . . . 4 ⊢ (𝐺 ∈ V → 𝐺 = ( I ‘𝐺)) |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (𝐺 ∈ V → (.g‘𝐺) = (.g‘( I ‘𝐺))) |
5 | fvprc 6097 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = ∅) | |
6 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
7 | 6 | fveq2d 6107 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (.g‘( I ‘𝐺)) = (.g‘∅)) |
8 | base0 15740 | . . . . . . . 8 ⊢ ∅ = (Base‘∅) | |
9 | eqid 2610 | . . . . . . . 8 ⊢ (.g‘∅) = (.g‘∅) | |
10 | 8, 9 | mulgfn 17367 | . . . . . . 7 ⊢ (.g‘∅) Fn (ℤ × ∅) |
11 | xp0 5471 | . . . . . . . 8 ⊢ (ℤ × ∅) = ∅ | |
12 | 11 | fneq2i 5900 | . . . . . . 7 ⊢ ((.g‘∅) Fn (ℤ × ∅) ↔ (.g‘∅) Fn ∅) |
13 | 10, 12 | mpbi 219 | . . . . . 6 ⊢ (.g‘∅) Fn ∅ |
14 | fn0 5924 | . . . . . 6 ⊢ ((.g‘∅) Fn ∅ ↔ (.g‘∅) = ∅) | |
15 | 13, 14 | mpbi 219 | . . . . 5 ⊢ (.g‘∅) = ∅ |
16 | 7, 15 | syl6eq 2660 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (.g‘( I ‘𝐺)) = ∅) |
17 | 5, 16 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝐺 ∈ V → (.g‘𝐺) = (.g‘( I ‘𝐺))) |
18 | 4, 17 | pm2.61i 175 | . 2 ⊢ (.g‘𝐺) = (.g‘( I ‘𝐺)) |
19 | 1, 18 | eqtri 2632 | 1 ⊢ · = (.g‘( I ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 I cid 4948 × cxp 5036 Fn wfn 5799 ‘cfv 5804 ℤcz 11254 .gcmg 17363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-neg 10148 df-z 11255 df-seq 12664 df-slot 15699 df-base 15700 df-mulg 17364 |
This theorem is referenced by: (None) |
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