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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovcom | Structured version Visualization version GIF version |
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7384. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
Ref | Expression |
---|---|
ndmaovcom | ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5576 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | |
2 | ndmaov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
3 | 2 | eqcomi 2743 | . . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹 |
4 | 3 | eleq2i 2825 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
5 | 1, 4 | bitr3i 280 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐴, 𝐵〉 ∈ dom 𝐹) |
6 | ndmaov 44301 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V) | |
7 | 5, 6 | sylnbi 333 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = V) |
8 | ancom 464 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
9 | opelxp 5576 | . . . 4 ⊢ (〈𝐵, 𝐴〉 ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
10 | 3 | eleq2i 2825 | . . . 4 ⊢ (〈𝐵, 𝐴〉 ∈ (𝑆 × 𝑆) ↔ 〈𝐵, 𝐴〉 ∈ dom 𝐹) |
11 | 8, 9, 10 | 3bitr2i 302 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ 〈𝐵, 𝐴〉 ∈ dom 𝐹) |
12 | ndmaov 44301 | . . 3 ⊢ (¬ 〈𝐵, 𝐴〉 ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V) | |
13 | 11, 12 | sylnbi 333 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐵𝐹𝐴)) = V) |
14 | 7, 13 | eqtr4d 2777 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 〈cop 4537 × cxp 5538 dom cdm 5540 ((caov 44236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-res 5552 df-iota 6327 df-fun 6371 df-fv 6377 df-aiota 44203 df-dfat 44237 df-afv 44238 df-aov 44239 |
This theorem is referenced by: (None) |
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