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Mirrors > Home > MPE Home > Th. List > muls0ord | Structured version Visualization version GIF version |
Description: If a surreal product is zero, one of its factors must be zero. (Contributed by Scott Fenton, 16-Apr-2025.) |
Ref | Expression |
---|---|
muls0ord.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
muls0ord.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
Ref | Expression |
---|---|
muls0ord | ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muls0ord.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ No ) | |
2 | muls02 28182 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s ) | |
3 | 1, 2 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s ) |
4 | 3 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ( 0s ·s 𝐵) = 0s ) |
5 | 4 | eqeq2d 2746 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ (𝐴 ·s 𝐵) = 0s )) |
6 | muls0ord.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No ) | |
7 | 6 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐴 ∈ No ) |
8 | 0sno 27886 | . . . . . . . . . 10 ⊢ 0s ∈ No | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 0s ∈ No ) |
10 | 1 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ∈ No ) |
11 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ≠ 0s ) | |
12 | 7, 9, 10, 11 | mulscan2d 28220 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ 𝐴 = 0s )) |
13 | 5, 12 | bitr3d 281 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s ↔ 𝐴 = 0s )) |
14 | 13 | biimpd 229 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s → 𝐴 = 0s )) |
15 | 14 | impancom 451 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐵 ≠ 0s → 𝐴 = 0s )) |
16 | 15 | necon1bd 2956 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (¬ 𝐴 = 0s → 𝐵 = 0s )) |
17 | 16 | orrd 863 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐴 = 0s ∨ 𝐵 = 0s )) |
18 | 17 | ex 412 | . 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s → (𝐴 = 0s ∨ 𝐵 = 0s ))) |
19 | oveq1 7438 | . . . . 5 ⊢ (𝐴 = 0s → (𝐴 ·s 𝐵) = ( 0s ·s 𝐵)) | |
20 | 19 | eqeq1d 2737 | . . . 4 ⊢ (𝐴 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ ( 0s ·s 𝐵) = 0s )) |
21 | 3, 20 | syl5ibrcom 247 | . . 3 ⊢ (𝜑 → (𝐴 = 0s → (𝐴 ·s 𝐵) = 0s )) |
22 | muls01 28153 | . . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s ) | |
23 | 6, 22 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s ) |
24 | oveq2 7439 | . . . . 5 ⊢ (𝐵 = 0s → (𝐴 ·s 𝐵) = (𝐴 ·s 0s )) | |
25 | 24 | eqeq1d 2737 | . . . 4 ⊢ (𝐵 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 ·s 0s ) = 0s )) |
26 | 23, 25 | syl5ibrcom 247 | . . 3 ⊢ (𝜑 → (𝐵 = 0s → (𝐴 ·s 𝐵) = 0s )) |
27 | 21, 26 | jaod 859 | . 2 ⊢ (𝜑 → ((𝐴 = 0s ∨ 𝐵 = 0s ) → (𝐴 ·s 𝐵) = 0s )) |
28 | 18, 27 | impbid 212 | 1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 No csur 27699 0s c0s 27882 ·s cmuls 28147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 df-subs 28069 df-muls 28148 |
This theorem is referenced by: mulsne0bd 28227 expsne0 28429 |
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