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Mirrors > Home > MPE Home > Th. List > sgrp2nmndlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for sgrp2nmnd 17240. (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 2632 | . . 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 2782 | . . . . . 6 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
6 | eqcom 2617 | . . . . . . . . 9 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
7 | eqeq2 2621 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵)) | |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵)) |
9 | 6, 8 | syl5rbbr 274 | . . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝐵 ↔ 𝑥 = 𝐴)) |
10 | 9 | notbid 307 | . . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴)) |
11 | 10 | biimpcd 238 | . . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
12 | 5, 11 | sylbi 206 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
13 | 12 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴)) |
14 | 13 | imp 444 | . . 3 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴) |
15 | 14 | iffalsed 4047 | . 2 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵) |
16 | simp2 1055 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆) | |
17 | simp1 1054 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑆) | |
18 | 4, 15, 16, 17, 16 | ovmpt2d 6686 | 1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ifcif 4036 {cpr 4127 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 +gcplusg 15768 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: sgrp2rid2 17236 sgrp2nmndlem4 17238 sgrp2nmndlem5 17239 |
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