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Mirrors > Home > MPE Home > Th. List > mgplem | Structured version Visualization version GIF version |
Description: Lemma for mgpbas 18318. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mgpbas.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
mgplem.2 | ⊢ 𝐸 = Slot 𝑁 |
mgplem.3 | ⊢ 𝑁 ∈ ℕ |
mgplem.4 | ⊢ 𝑁 ≠ 2 |
Ref | Expression |
---|---|
mgplem | ⊢ (𝐸‘𝑅) = (𝐸‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgplem.2 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
2 | mgplem.3 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 15716 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | mgplem.4 | . . . 4 ⊢ 𝑁 ≠ 2 | |
5 | 1, 2 | ndxarg 15715 | . . . . 5 ⊢ (𝐸‘ndx) = 𝑁 |
6 | plusgndx 15803 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
7 | 5, 6 | neeq12i 2848 | . . . 4 ⊢ ((𝐸‘ndx) ≠ (+g‘ndx) ↔ 𝑁 ≠ 2) |
8 | 4, 7 | mpbir 220 | . . 3 ⊢ (𝐸‘ndx) ≠ (+g‘ndx) |
9 | 3, 8 | setsnid 15743 | . 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉)) |
10 | mgpbas.1 | . . . 4 ⊢ 𝑀 = (mulGrp‘𝑅) | |
11 | eqid 2610 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
12 | 10, 11 | mgpval 18315 | . . 3 ⊢ 𝑀 = (𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉) |
13 | 12 | fveq2i 6106 | . 2 ⊢ (𝐸‘𝑀) = (𝐸‘(𝑅 sSet 〈(+g‘ndx), (.r‘𝑅)〉)) |
14 | 9, 13 | eqtr4i 2635 | 1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ℕcn 10897 2c2 10947 ndxcnx 15692 sSet csts 15693 Slot cslot 15694 +gcplusg 15768 .rcmulr 15769 mulGrpcmgp 18312 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
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-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-sets 15701 df-plusg 15781 df-mgp 18313 |
This theorem is referenced by: mgpbas 18318 mgpsca 18319 mgptset 18320 mgpds 18322 |
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