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Mirrors > Home > MPE Home > Th. List > sgrp2nmndlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for sgrp2nmnd 18740. (Contributed by AV, 29-Jan-2020.) |
Ref | Expression |
---|---|
mgm2nsgrp.s | ⊢ 𝑆 = {𝐴, 𝐵} |
mgm2nsgrp.b | ⊢ (Base‘𝑀) = 𝑆 |
sgrp2nmnd.o | ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) |
sgrp2nmnd.p | ⊢ ⚬ = (+g‘𝑀) |
Ref | Expression |
---|---|
sgrp2nmndlem3 | ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgrp2nmnd.p | . . . 4 ⊢ ⚬ = (+g‘𝑀) | |
2 | sgrp2nmnd.o | . . . 4 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) | |
3 | 1, 2 | eqtri 2764 | . . 3 ⊢ ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) |
4 | 3 | a1i 11 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))) |
5 | df-ne 2944 | . . . . . 6 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
6 | eqeq2 2748 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵)) | |
7 | 6 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵)) |
8 | eqcom 2743 | . . . . . . . . 9 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
9 | 7, 8 | bitr3di 285 | . . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝐵 ↔ 𝑥 = 𝐴)) |
10 | 9 | notbid 317 | . . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴)) |
11 | 10 | biimpcd 248 | . . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
12 | 5, 11 | sylbi 216 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
13 | 12 | 3ad2ant3 1135 | . . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
14 | 13 | imp 407 | . . 3 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴) |
15 | 14 | iffalsed 4497 | . 2 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵) |
16 | simp2 1137 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆) | |
17 | simp1 1136 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑆) | |
18 | 4, 15, 16, 17, 16 | ovmpod 7507 | 1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ifcif 4486 {cpr 4588 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Basecbs 17083 +gcplusg 17133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 |
This theorem is referenced by: sgrp2rid2 18736 sgrp2nmndlem4 18738 sgrp2nmndlem5 18739 |
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