Theorem divmulsw 28423

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28448, we can eliminate the existence hypothesis (see divmuls 28451). (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
divmulsw	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem divmulsw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divsval 28419	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → (𝐴 /su 𝐶) = (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴))
2	1	eqeq1d 2768	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
3	2	3expb 1138	. . . 4 ⊢ ((𝐴 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
4	3	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
5	4	adantr 486	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
6		simpl2 1211	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → 𝐵 ∈ No )
7		simp3l 1220	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ∈ No )
8		simp3r 1221	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ≠ 0s )
9		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐴 ∈ No )
10	7, 8, 9	3jca 1146	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → (𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ))
11		noreceuw 28421	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
12	10, 11	sylan 592	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
13		oveq2 7431	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 ·s 𝑦) = (𝐶 ·s 𝐵))
14	13	eqeq1d 2768	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐶 ·s 𝑦) = 𝐴 ↔ (𝐶 ·s 𝐵) = 𝐴))
15	14	riota2 7405	. . 3 ⊢ ((𝐵 ∈ No ∧ ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
16	6, 12, 15	syl2anc 596	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
17	5, 16	bitr4d 285	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∃wrex 3092  ∃!wreu 3370  ℩crio 7379  (class class class)co 7423   No csur 27841   0_s c0s 28035   1_s c1s 28036   ·_s cmuls 28336   /_su cdivs 28417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337  df-divs 28418

This theorem is used by:  divmulswd  28424  divs1  28434  divmuls  28451